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\begin{document}

\title[]{Extension of smooth functions in infinite dimensions II: manifolds}
\author[]{C. J. Atkin}
\address{School of Mathematical and Computing Sciences, Victoria University
of Wellington, P. O. Box 600, Wellington, New Zealand}
%\curraddr{}
\email{chris.atkin@vuw.ac.nz}
%\dedicatory{}
\date{December 2001}
%\thanks{}
% \translator{}
%\keywords{}
\subjclass{46T20}
\begin{abstract}
Let $M$ be a separable C$^\infty$ Finsler manifold of infinite dimension.
Then it is proved, amongst other results, \ that under suitable conditions of
local extensibility the germ of a C$^\infty$ function, or of a C$^\infty$
section of a vector bundle, on the union of a closed submanifold and a closed
locally compact set in $M$, extends to a C$^\infty$ function on the whole of
$M$. 
\end{abstract}
\maketitle

\section{Introduction.}\label{secta}

In the preceding paper \cite{atkin2}, I presented a theorem,
\cite{atkin2}5.1, which asserts the possibility of extending functions of
various classes from a finite union of convex sets to the whole of a locally
convex space. A particular example, assuming the space to be normed, would be
the Fr\a'echet C$^\infty$ differentiability class. The purpose of this paper
is to show how that theorem may be used to obtain extension theorems for
functions of a given class from suitable subsets of a manifold to the whole
manifold. 

\label{para2}An extension theorem will say that any ``function'' of a certain
class defined on a subset $X$ of a space $M$ is necessarily the
``restriction'' of a ``function'' of some related class on $M$. The
``functions'' need not always be functions in the usual sense, and
``restriction'' may not be a completely obvious procedure, as with the trace
of a Sobolev function on the boundary of its domain. Setting aside such
possibilities, there are several kinds of extension theorem that one might
consider. One might ask simply for a function on $M$ whose restriction to $X$
is given; or one might, for instance, also ask that its infinitesimal
behaviour at points of $X$ should be prescribed. This is the content of
Whitney's famous theorem, \cite{whitney}, which in effect extends suitable
C$^\infty$-jets on a closed set $X$ in$\,\,\mathbb{R}^n\,$ to C$^\infty$
functions on $\,\mathbb{R}^n$. One might also ask for a function on $M$ whose
\emph{germ }at $X$ agrees with a given germ, that is, it agrees on some
unspecified small neighbourhood of $X$ with a given function on some
neighbourhood of $X$. The principal theorem presented here, \ref{theorem}, is
of this last kind: if $X$ is a suitable subset (a closed locally compact
subset, a closed submanifold, or a union of both) of the manifold $M$, a
``smooth'' function $f$ defined on a neighbourhood $V$ of $X$ coincides on a
certain smaller neighbourhood $W$ of $X$ with a ``smooth'' function on $M$.
The neighbourhood $W$ depends only on $V$ and $X$. 

When $X$ is just a pair of points, \ref{theorem} has the following very weak
but interesting consequence, proved as \ref{end} below:

\begin{lemma}
\ Let $M$ be a paracompact \emph{C}$^\infty$\emph{ }manifold modelled on the
Banach space $C$ or on the Banach space $\,\ell^1\,$. The class of
real-valued \emph{C}$^\infty$\emph{ }functions on $M$ separates points.
\end{lemma}

This answers the question raised by Elworthy in \cite{elworthy}, and
mentioned in the introduction to \cite{atkin2}. In fact, the restriction to
$C$ and $\,\ell^1\,$ as models is quite unnecessary, in view of extension
properties and smoothing theorems that will be presented in later papers.

For the sake of general applicability, my principal results are presented in
terms of the extension of sections of fibrations rather than of functions.
There are theorems similar to \ref{theorem} that may be derived from
\cite{atkin2}5.5; I have omitted them, since their proofs are essentially the
same, and their hypotheses inevitably more complicated. Some of the
definitions of \cite{atkin2}\S 2 will be assumed.

\ 

There are two obvious difficulties in applying the results of \cite{atkin2}
on manifolds. \cite{atkin2}5.1 and its variants depend on convexity (see
\cite{atkin2}3.3 for some remarks on this) and on ``extensibility'' of the
functions considered (see \cite{atkin2}1.5 for the definition). One cannot
usually expect either of these properties to be preserved by the coordinate
transformations of a manifold. However, for a C$^1$ manifold modelled on a
separable normed space with the Fr\a'echet definition of differentiability,
the charts may be so chosen that the coordinate transformations preserve
convexity ``approximately'', and this is enough for the proof here. The
requirement that the functions in question be extensible is not so simply
overcome, but, as I pointed out at \cite{atkin2}1.3, there are many standard
models for which the C$^\infty$ class, at least, consists entirely of
extensible functions. As I have already hinted, I hope to demonstrate
elsewhere that, for many cases of interest, the requirement of extensibility
either is satisfied automatically or can be satisfied by a careful choice of
charts.

When the model is not a normed space, there is no natural notion of
differentiability, and the definitions usually presented (see
\cite{frolicher} and \cite{michor}, for example) are insufficient to support
the ``approximate convexity'' arguments used in the normed case. Thus the
result that I present for such models, Theorem \ref{badjob}, is much weaker,
but not altogether uninteresting. 

\ \ 

Whitney's method of proving his extension theorem cannot be employed in
infinite dimensions, since it employs smooth partitions of unity, which need
not always exist (\cite{bonicframpton}, \cite{devillegodefroyzizl}), and
covers by bounded cubes with disjoint interiors, which never can. There are,
in fact, well-known counterexamples to many natural conjectures. If $f$ is a
function which is $1$ on a neighbourhood of the origin of a separable normed
space $E$ of infinite dimension and is $0$ on a neighbourhood of the unit
sphere $S$ of $E$, any C$^\infty$ extension of $f$ to the whole of $E$ may
easily be adjusted to give a non-zero C$^\infty$ function of bounded support,
which, by \cite{bonicframpton} or \cite{devillegodefroyzizl}, is impossible
in many spaces (such as $C$ and $\,\ell^1\,$). 

At first glance, the reason for this counterexample is that $S$ separates
$E$. However, the set $\,S\times\{0\}\,$ in $\,E\times E\,$ is
homotopy-negligible (the inclusion
\begin{equation*}
(E\times E)\setminus(S\times\{0\})\longrightarrow E\times E
\end{equation*}
is a homotopy equivalence), and yet any C$^\infty$ extension to $\,E\times
E\,$ of a function which is $1$ at $\,(0,0)\,$ and $0$ on $\,S\times\{0\}\,$
would also lead to a non-zero C$^\infty$ function of bounded support in
$\,E\times\{0\}\,$. In these examples, it is immaterial that $S$ is a
norm-sphere; any bounded closed set separating a subspace of $E$ that does
not admit C$^\infty$ non-zero functions of bounded support would give a like
result. It seems, then, that certain sets in some spaces are ``bad'' for
extension problems, and that their ``badness'' is a rather subtle property,
related not merely to their topological embedding in the ambient space but
also to its analytical or geometrical character. There is a vague analogy
with the theory of several complex variables, in which spheres are similarly
``bad'' for holomorphic functions. One cannot expect a simple
characterization of ``good'' sets for a given extension problem, since it
would have to depend on the spaces and the function classes involved;
nevertheless, the theorems of this paper do assert the ``goodness'', for the
problems considered, of certain rather simple kinds of set.

\ \ 

The arguments here were developed over a very long period. I am grateful to
the Institutes of Mathematics of the University of Warsaw and of the Polish
Academy of Sciences for the hospitality which enabled me at last to put them
into a reasonable form, and to the referee for helpful suggestions. 

\section{Localization and fibrations.}

\begin{definition}
Su\label{localcat}ppose that $\mathcal{C}$ is a category of mappings between
topological spaces. (There may be very few objects in the category
$\mathcal{C}$). Extend $\mathcal{C}$ by ``localization'' to the class
$\mathcal{L}\mathcal{C}$ of \emph{locally-}$\mathcal{C}$\emph{ mappings
}between open sets of objects of $\mathcal{C}$, as follows.

Let $U$ be an open set in $X$ and $V$ an open set in $Y$, where both $X$ and
$Y$ are objects of $\mathcal{C}$, and let $\;i:V\longrightarrow Y\;$ denote
the inclusion. Then $\;f:U\longrightarrow V\;$ belongs to
$\mathcal{L}\mathcal{C}$ if and only if, for each $\;x\in U\,$, there exist a
neighbourhood $W$ of $x$ in $U$ and a morphism $\;g:X\longrightarrow Y\;$ of
$\mathcal{C}$ such that $\;g\mid W=i\circ f\mid W\,$. 
\end{definition}

In the language of \cite{atkin2}1.5, $\mathcal{L}\mathcal{C}$ consists of the
$\mathcal{C}$-extensible mappings of open sets of objects of $\mathcal{C}$.
If $\mathcal{C}$ consists of continuous mappings, then so does
$\mathcal{L}\mathcal{C}$, which is itself a category. The mappings in
$\mathcal{L}\mathcal{C}$ are automatically locally defined (see
\cite{atkin2}1.4). 

\begin{definition}
\ T\label{admiss}he category $\mathcal{C}$ is \emph{admissible }if its
objects are topological vector spaces and, for each pair $\,E,F\,$ of objects
of $\mathcal{C}$, the $\mathcal{C}$-morphisms from $E$ to $F$ form an
$F$-admissible class (see \cite{atkin2}1.7). \ 
\end{definition}

It then follows that, if $X$ is open in $E$ and $Y$ is open and convex in
$F$, the $\mathcal{L}\mathcal{C}$-morphisms $\;X\longrightarrow Y\;$ are
$Y$-admissible in the sense of \cite{atkin2}1.7.

\begin{definition}
Giv\label{lcman}en a category $\mathcal{C}$, as in \ref{localcat} above,
which consists of continuous maps, \emph{an
}$\mathcal{L}\mathcal{C}$\emph{-manifold }is a manifold, modelled on an
object $E$ of $\mathcal{C}$, for which all the coordinate transformations are
isomorphisms in $\mathcal{L}\mathcal{C}$. \ \ \ 
\end{definition}

In other words, $\mathcal{L}\mathcal{C}$ determines a pseudogroup
$\,\mathfrak{G}_E(\mathcal{C})\,$ of transformations in $E$, consisting of
the mappings $\;f:U\longrightarrow V\;$ between open subsets of $E$ which are
in $\mathcal{L}\mathcal{C}$ and have inverses $\;f^{-1}:V\longrightarrow U\;$
also in $\mathcal{L}\mathcal{C}$. An $\mathcal{L}\mathcal{C}$-manifold
modelled on $E$ is then a $\,\mathfrak{G}_E(\mathcal{C})$-manifold. I shall
always write of $\mathcal{L}\mathcal{C}$-manifolds and
$\mathcal{L}\mathcal{C}$-charts, rather than mention the pseudogroup
$\,\mathfrak{G}_E(\mathcal{C})\,$; this accords with familiar usage. 

The category $\mathcal{L}\mathcal{C}$ may now be further extended. If $M$ and
$N$ are $\mathcal{L}\mathcal{C}$-manifolds, modelled respectively on the
objects $E$ and $F$ of $\mathcal{C}$, a mapping $\;f:M\longrightarrow N\;$ is
described as being in $\mathcal{L}\mathcal{C}$, or as being an
$\mathcal{L}\mathcal{C}$-mapping, if, for each $\;x\in M\,$, there are open
neighbourhoods $U$ of $x$ in $M$ and $V$ of $\,f(x)\,$ in $N$, and
$\mathcal{L}\mathcal{C}$-manifold charts
$\;\phi:U\longrightarrow\phi(U)\subseteq E\,$,
$\;\psi:V\longrightarrow\psi(V)\subseteq F\,$, where $\,\phi(U)\,$ is open in
$E$ and $\,\psi(V)\,$ is open in $F$, such that $\;f(U)\subseteq V\;$ and
$\;\psi\circ(f\mid U)\circ\phi^{-1}:\phi(U)\longrightarrow\psi(V)\;$ is in
$\mathcal{L}\mathcal{C}$. 

If this condition holds for one specific choice of
$\mathcal{L}\mathcal{C}$-manifold charts $\phi$ and $\psi$ with the given
domains $U$ and $V$, it holds also for all other choices. Since $\mathcal{C}$
is assumed to consist of continuous maps, an $\mathcal{L}\mathcal{C}$-mapping
of $\mathcal{L}\mathcal{C}$-manifolds must be continuous; and such mappings
form a category, which may be called the $\mathcal{L}\mathcal{C}$-category.
(This name is somewhat ambiguous, in view of \ref{localcat}, but it is an
ambiguity of a kind familiar from the C$^p$ category). 

An open subset $U$ of an $\mathcal{L}\mathcal{C}$-manifold $M$ itself clearly
has a unique $\mathcal{L}\mathcal{C}$-manifold structure such that the
inclusion $\;U\longrightarrow M\;$ is an $\mathcal{L}\mathcal{C}$-mapping.
Furthermore, any $\mathcal{L}\mathcal{C}$-mapping into $M$ whose image is
included in $U$ is then an $\mathcal{L}\mathcal{C}$-mapping into $U$. 

\begin{definition}
\ A \label{applic}category $\mathcal{D}$ is \emph{applicable} if its objects
are open sets in topological vector spaces, its morphisms are continuous
mappings, and, together with any morphism $\phi$, it contains all mappings
obtained from $\phi$ by restriction to an open subset of its domain or image.
(The categories $\mathcal{L}\mathcal{C}$ are examples of applicable
categories). 
\end{definition}


\begin{definition}
\ Su\label{subman}ppose that the topological vector space $E$ is an object of
the applicable category $\mathcal{D}$, and $F$ is a linear subspace of $E$.
Let $M$ be a $\mathcal{D}$-manifold modelled on $E$. A
$\mathcal{D}$\emph{-submanifold of }$M$\emph{ modelled on }$F$\emph{ }is a
non-null subset $N$ of $M$ which may be covered by $\mathcal{D}$-charts
$\;\phi:U\longrightarrow\phi(U)\subseteq E\;$ of $M$ such that $\;\phi(U\cap
N)=\phi(U)\cap F\,$. Such charts may be called
$\mathcal{D}$-\emph{submanifold charts} for $N$.

When $E$ is a normed space and $M$ is a C$^1$-manifold modelled on $E$, a
C$^1$\emph{-submanifold }modelled on $F$ is, similarly, a non-null subset $N$
of $M$ such that every point of $M$ is contained in the domain of some C$^1$
chart of $M$, $\;\phi:U\longrightarrow\phi(U)\subseteq E\,$, for which \
$\;\phi(U\cap N)=\phi(U)\cap F\,.\,$
\end{definition}

The submanifolds just defined are the only ones that will be considered in
this paper, although in other contexts other definitions are appropriate. In
this case, the submanifold $N$ is closed in $M$ if and only if its model $F$
is closed in $E$. 


\begin{remark}
\ Th\label{submanrem}e $\mathcal{L}\mathcal{C}$-category is merely an ad hoc
tool for the convenient formulation of theorems, and is not a useful general
setting for geometry, even when $\mathcal{C}$ is the C$^\infty$ category; for
it is not, in principle, closed under simple analytical operations such as
the solution of ordinary differential equations. Of course, if one restricts
attention to many ``standard'' normed spaces,
$\;\mathcal{L}$C$^\infty=$C$^\infty\;$ (see \cite{atkin2}1.3), and there is
no problem. 

As defined here, an $\mathcal{L}\mathcal{C}$-submanifold need not even be an
$\mathcal{L}\mathcal{C}$-manifold in a natural way, although a
C$^1$-submanifold of a C$^1$-manifold does have a natural C$^1$ structure and
the same is true for many other common choices of $\mathcal{C}$. 
\end{remark}

\begin{definition}
\ Gi\label{lcimbedded}ven the category $\mathcal{C}$ as in \ref{localcat} and
an object $E$ of $\mathcal{C}$, a subspace $F$ of $E$ is
$\mathcal{C}$\emph{-imbedded} if the inclusion $\;i:F\longrightarrow E\;$
belongs to $\mathcal{C}$ and any mapping $\;g:H\longrightarrow F\;$ (where
$H$ is an object of $\mathcal{C}$) such that $\;i\circ g\in\mathcal{C}\;$
itself belongs to $\mathcal{C}$. 
\end{definition}

\begin{lemma}
\ Le\label{splitsub}t $N$ be an $\mathcal{L}\mathcal{C}$-submanifold,
modelled on the subspace $F$, of the $\mathcal{L}\mathcal{C}$-manifold $M$
modelled on the space $E$, and suppose that $F$ is $\mathcal{C}$-imbedded in
$E$. Then there is a unique $\mathcal{L}\mathcal{C}$-structure on $N$ such
that the inclusion $\,N\longrightarrow M\;$ is an
$\mathcal{L}\mathcal{C}$-mapping.
\end{lemma}

\begin{proof}
If $\;\phi:U\longrightarrow\phi(U)\subseteq E\,\;$ and
$\;\psi:V\longrightarrow\psi(V)\subseteq E\;$ are
$\mathcal{L}\mathcal{C}$-charts of $M$ such that $\;\phi(U\cap N)=\phi(U)\cap
F\;$ and $\;\psi(V\cap N)=\psi(V)\cap F\,$, then the coordinate
transformation 
\begin{equation*}
\,\chi:=(\phi\mid U\cap V)\circ(\psi\mid U\cap V)^{-1}:\psi(U\cap
V)\longrightarrow\phi(U\cap V)\,
\end{equation*}
in $\mathcal{L}\mathcal{C}$ carries $\,\psi(U\cap V)\cap F\,$ into
$\;\phi(U\cap V)\cap F\,$. However, if $F$ is $\mathcal{C}$-imbedded in $E$,
this implies that $\,\chi\mid\psi(U\cap V)\cap F\,$ as a mapping into
$\;\phi(U\cap V)\cap F\,$ is itself in $\mathcal{L}\mathcal{C}$. The
$\mathcal{L}\mathcal{C}$-structure on $N$ thus defined is the only possible
one making the inclusion $\mathcal{L}\mathcal{C}$. \ 
\end{proof}

\begin{definition}
\ Le\label{product}t $\,E,F\,$ be objects of $\mathcal{C}$; suppose that,
whenever mappings $\;\phi:E\longrightarrow E\;$ and $\;\psi:F\longrightarrow
F\;$ are in $\mathcal{C}$, $\;\phi\times\psi:E\times F\longrightarrow E\times
F\;$ is also in $\mathcal{C}$. Then, for any two
$\mathcal{L}\mathcal{C}$-manifolds $\,M,N\,$ with models $\,E,F\,$
respectively, there is an $\mathcal{L}\mathcal{C}$-manifold $\,M\times N\,$
defined by an atlas whose charts are the products of charts for $M$ and for
$N$ in the usual way.

Assume further that the projections $\;E\times F\longrightarrow E\;$ and
$\;E\times F\longrightarrow F\,$, and the diagonal map $\;E\longrightarrow
E\times E\,$, are in $\mathcal{C}$ (compare \ref{splitsub}); then both the
coordinate projections $\;p_1:M\times N\longrightarrow M\;$ and
$\;p_2:M\times N\longrightarrow N\,$, and the diagonal map
$\;\nabla:M\longrightarrow M\times M\,$, are also in $\mathcal{C}$.

By a \emph{locally trivial}$\;\mathcal{L}\mathcal{C}$\emph{-fibration }(with
base $M$, whose model is $E$, and fibre $N$, whose model is $F$) I mean an
$\mathcal{L}\mathcal{C}$-mapping $\;\pi:\mathcal{E}\longrightarrow M\;$ of
$\mathcal{L}\mathcal{C}$-manifolds, such that, for each $\;x\in M\,$, there
are an open neighbourhood $U$ of $x$ in $M$ (hence $U$ and $\,\pi^{-1}(U)\,$
have natural $\mathcal{L}\mathcal{C}$-structures, by the remarks after
\ref{lcman}) and an $\mathcal{L}\mathcal{C}$-isomorphism
$\;\phi:\pi^{-1}(U)\longrightarrow U\times N\;$ for which
$\;p_1\circ\phi=\pi\,$. Such a neighbourhood $U$ is called
an$\;\mathcal{L}\mathcal{C}$\emph{-trivializing neighbourhood }(of $x$ in
$M$, for the fibration $\pi$), and such an
$\mathcal{L}\mathcal{C}$-diffeomorphism $\phi$ may be called an
$\mathcal{L}\mathcal{C}$\emph{-local trivialization }for $\pi$ over the
\emph{base} $U$. Any smaller open neighbourhood of $x$ will then also be
$\mathcal{L}\mathcal{C}$-trivializing, an $\mathcal{L}\mathcal{C}$-local
trivialization being furnished by the restriction of $\phi$. The notion of
the \emph{restriction} of the locally trivial
$\mathcal{L}\mathcal{C}$-fibration $\pi$, also called $\mathcal{E}$, to a
(suitable --- for instance open) subset $V$ of $M$ is defined as usual, and
will be denoted by $\,\mathcal{E}\mid V\,$. 

By an $\mathcal{L}\mathcal{C}$-\emph{vector fibration} (with base $M$ and
fibre $E$) I mean a locally trivial $\mathcal{L}\mathcal{C}$-fibration
$\;\pi:\mathcal{E}\longrightarrow M\;$ for which each fibre is given the
structure of a topological vector space isomorphic to $E$, in such a way that
$M$ may be covered by the bases $U$ of $\mathcal{L}\mathcal{C}$-local
trivializations $\;\phi:\pi^{-1}(U)\longrightarrow U\times E\;$ whose
restriction to each fibre is a topological linear isomorphism. These may be
called $\mathcal{L}\mathcal{C}$-local vector trivializations of $\pi$.
\end{definition}

Equivalently, the transformation $\,\psi\circ\phi^{-1}\,$ between two such
local trivializations is to be a topological linear isomorphism in each
fibre. This is a weaker definition than is usually given (see \cite{lang})
for a vector \emph{bundle }with normed infinite-dimensional fibres. For a
C$^0$-vector bundle whose fibre is a normed space $E$, the bundle transition
functions $\;\theta_{UV}:U\cap V\longrightarrow GL(E)\;$ between two local
bundle trivializations $\;\phi:\pi^{-1}(U)\longrightarrow U\times E\;$ and
$\;\psi:\pi^{-1}(V)\longrightarrow V\times E\;$ are defined by setting
$\;\theta_{UV}(x)\xi:=\psi\circ\phi^{-1}(x,\xi)\;$ for any $\;x\in U\cap V\;$
and $\;\xi\in E\,$, and are required to be continuous as maps into
$\,GL(E)\,$ with the uniform norm topology. This condition is evidently
needed if the notion of a Finsler structure in the bundle is to make sense.
(For clarity, I shall call such a Finsler structure a \emph{bundle} Finsler
structure; a Finsler structure on a manifold, which is the same as a bundle
Finsler structure in its tangent bundle, may be called a \emph{tangential}
Finsler structure). A C$^1$-vector fibration is a C$^0$-vector bundle, since
the transition functions may be obtained by differentiation at the points of
the zero-section; but it is also a C$^1$-manifold, which a C$^0$-vector
bundle need not be. This is the ``loss of differentiability problem''
discussed in \cite{elworthy}.

\begin{definition}
\ Le\label{subfib}t $\;\pi:\mathcal{E}\longrightarrow M\;$ be an
$\mathcal{L}\mathcal{C}$-vector fibration with base $M$ and fibre $E$, and
suppose that $\,\mathcal{E}_1\,$ is a subset of $\mathcal{E}$ and $G$ is an
$\mathcal{C}$-imbedded vector subspace of $E$. Then
$\;\pi_1:=\pi\mid\mathcal{E}_1\;$ is an $\mathcal{L}\mathcal{C}$\emph{-vector
subfibration} of $\pi$, with fibre $G$, if \ $M$ may be covered by the bases
$U$ of $\mathcal{L}\mathcal{C}$-local trivializations
$\;\phi:\pi^{-1}(U)\longrightarrow U\times E\;$ such that 
\begin{equation*}
\phi(\pi^{-1}(U)\cap\mathcal{E}_1)=U\times G\,.
\end{equation*}
In such a case, $\,\pi_1\,$ is an $\mathcal{L}\mathcal{C}$-vector fibration
in its own right (as $G$ is $\mathcal{C}$-imbedded in $E$).

Provided that the category $\mathcal{C}$ fulfils suitable conditions on the
formation of quotients, one may construct a \emph{quotient
}$\mathcal{L}\mathcal{C}$\emph{-vector fibration}$\,\;\pi/\pi_1\,$ (or
$\,\mathcal{E}/\mathcal{E}_1\,$), which is an $\mathcal{L}\mathcal{C}$-vector
fibration with base $M$ and fibre $\,E/G\,$. (The category C$^0$ is certainly
satisfactory for this purpose). Similarly, an
$\mathcal{L}\mathcal{C}$\emph{-morphism }of $\mathcal{L}\mathcal{C}$-vector
fibrations is an $\mathcal{L}\mathcal{C}$-map which respects fibres and is
linear in each fibre; the inclusion map of an $\mathcal{L}\mathcal{C}$-vector
subfibration, and the quotient map when it is defined, are
$\mathcal{L}\mathcal{C}$-morphisms. $\mathcal{L}\mathcal{C}$-morphisms may be
added and multiplied by scalars. There are similar definitions for
C$^0$-vector bundles.
\end{definition}


\begin{definition}
\ L\label{lcsection}et $\;\pi:L\longrightarrow M\;$ be a locally trivial
$\mathcal{L}\mathcal{C}$-fibration, as above. A \emph{section }of $\pi$ over
a subset $X$ of $M$ is of course a function $\;\sigma:X\longrightarrow L\;$
such that $\,\pi\circ\sigma\,$ is the inclusion of $X$ in $M$. When $U$ is an
open subset of $M$, an $\mathcal{L}\mathcal{C}$\emph{-section} of $\pi$ over
$U$ is a section which is an$\;\mathcal{L}\mathcal{C}$-mapping.
\end{definition}

An $\mathcal{L}\mathcal{C}$-local trivialization
$\;\phi:\pi^{-1}(U)\longrightarrow U\times N\;$ establishes a one-one
correspondence between $\mathcal{L}\mathcal{C}$-sections of $\pi$ over $U$
and $\mathcal{L}\mathcal{C}$-mappings $\;U\longrightarrow N\,$; the section
$\sigma$ defines the mapping $\,p_2\circ\phi\circ\sigma\,$, and the mapping
$f$ gives the section $\sigma$, where $\;\sigma(x):=\phi^{-1}\circ(1\times
f)\circ\nabla(x)\,$.\\
\ 

\begin{remark}
\ Th\label{pullback}e usual constructions, such as pullbacks, may be carried
out for locally trivial $\mathcal{L}\mathcal{C}$-fibrations, provided that
$\mathcal{C}$ is large enough for the constructions to make sense. Therefore,
at \ref{prop}, I shall restrict discussion to
$\mathcal{L}\mathcal{C}$-sections of a locally trivial
$\mathcal{L}\mathcal{C}$-fibration $\;\pi:L\longrightarrow M\,$. There are
similar results for $\mathcal{L}\mathcal{C}$-liftings to $L$ of an
$\mathcal{L}\mathcal{C}$-mapping $\;k:R\longrightarrow M\,$, where $R$ is
another $\mathcal{L}\mathcal{C}$-manifold; they may be proved either by the
same method, with some slight complications of notation, or, if $\mathcal{C}$
is a suitable category, by associating such liftings with sections of
$\;k^\ast\pi\,$. 
\end{remark}

\ 

\section{An extension theorem.}

This section is devoted to the proof of the technical result \ref{prop},
whose application occupies the rest of the paper. The proof is inductive; its
structure is very similar to the proof of \cite{atkin2}4.2, and it depends on
\cite{atkin2}5.1. In particular, the definitions of \cite{atkin2}\S 2 will be
used without explanation. $\mathcal{D}$ is to denote an applicable category
(see \ref{applic}), and the convex envelope of a subset $Q$ in a vector space
$E$ will be denoted by $\,$co$(Q)\,$.

\begin{definition}
\ L\label{extfam}et $M$ be a $\mathcal{D}$-manifold. An \emph{extension
family }in $M$ is a pair $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ consisting of
a class of pairs of open sets indexed by a finite or countable set $J$, and
of a subset $\Theta$ of $J$, which satisfies the conditions: \\
\hspace*{0.25cm}%
\begin{tabular}[t]{ll}
\emph{(i)}&the class $\;\{U_0^j:j\in J\}\;$ covers $M$;\\
\emph{(ii)}&for any $\;\psi\in J\setminus\Theta\,$, the set
$\;\Theta_\psi:=\{\theta\in\Theta:U^\psi\cap U^\theta\neq\emptyset\}\;$ is
finite;\\
\emph{(iii)}&for any $\;j\in J\,$, $\;$cl$_M(U_0^j)\subseteq U^j\,$.
\end{tabular}%
\\
\label{correction1}For any extension family, I shall write 
\begin{equation*}
U_0(\Theta):=\bigcup_{i\in\Theta}U_0^i\quad\text{and}\quad
U(\Theta):=\bigcup_{i\in\Theta}U^i\,.
\end{equation*}
\ \ 
\end{definition}

\begin{lemma}
\ I\label{closed}f $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ is an extension
family in $M$, then $\;$\emph{cl}$_M(U_0(\Theta))\subseteq U(\Theta)\,$.
\end{lemma}

\begin{proof}
Suppose $\;x\in$cl$_M(U_0(\Theta))\setminus U_0(\Theta)\,$. By
\ref{extfam}\emph{(i)}, there is some $\;\psi\in J\setminus\Theta\;$ such
that $\;x\in U_0^\psi\,$, and $\,U^\psi\,$ meets $\,U_0^\theta\,$ (for
$\;\theta\in\Theta\,$) only when $\theta$ is in the finite set
$\,\Theta_\psi\,$. Hence,
$\;x\in$cl$_M\left(\bigcup_{\theta\in\Theta_\psi}U_0^\theta\right)=\bigcup_{%
\theta\in\Theta_\psi}$cl$_M(U_0^\theta)\,$, and
$\,\;$cl$_M(U_0^\theta)\subseteq U^\theta\subseteq U(\Theta)\;$ for
$\;\theta\in\Theta\,$. 
\end{proof}

\begin{definition}
\ If\label{extsys} $M$ is a $\mathcal{D}$-manifold and $\mathfrak{V}$ is an
open cover of $M$, a $\mathfrak{V}$-\emph{extension system} in $M$ is an
extension family $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ satisfying the
additional condition\\
\hspace*{0.5cm}\emph{(iv)}\hspace*{0.5cm}for any finite subset $\;T\subseteq
J\;$ and any $\;\psi\in J\setminus\Theta\,$, there is a$\,$
$\mathcal{D}$-chart $\;\omega_{\psi,T}:O_{\psi,T}\longrightarrow E\;$ for $M$
such that\\
\hspace*{1.5cm}\emph{(a)\label{m1}}\hspace*{0.5cm}$O_{\psi,T}\subseteq V\;$
for some $\;V\in\mathfrak{V}\,$,\\
\hspace*{1.5cm}\emph{(b)}\hspace*{0.5cm}when $\;k,l\in T\;$ and $\;U^\psi\cap
U^k\neq\emptyset\neq U^k\cap U^l\,$, then $\;U^l\subseteq
O_{\psi,T}\;\label{someclosure}$ and
$\;$co$(\omega_{\psi,T}(U_0^l))\prec\omega_{\psi,T}(U^l)\,$.

If $\mathfrak{V}$ consists of all the open sets of $M$, one might speak of an
\emph{extension system}.
\end{definition}

If one takes $\;T:=\{j\}\,$, \emph{(iv)(b)} shows that $\,\{U^j\}\,$ must
refine $\mathfrak{V}$. 

\begin{remark}
\ I\label{comments}n most respects, the above definition \ref{extsys} is what
is needed in the proof of \ref{prop}. However, the full strength of
\emph{(iv)} is superfluous, for $J$ is taken in \ref{prop} to be a subset of
$\mathbb{N}$ and, for a given $\;\psi\in J\setminus\Theta\,$, the only set
$T$ that is used is $\,(J\cap[1,\psi])\cup\left(\bigcup_{j\in
J\cap[1,\psi]}\Theta_j\right)\,$ (so that the chart $\,\omega_{\psi,T}\,$
depends only on $\psi$). \ref{extsys} could, therefore, be weakened somewhat,
but at the cost of assuming a specific ordering of $J$. 

The details of the argument are somewhat arbitrary, and, in fact, to apply
\ref{prop}, I shall prove the substantially stronger properties \ref{strong}.
As a definition, therefore, \ref{extsys} is an ugly compromise; but it has
the potential advantage that, as in the suggested weaker version,
\emph{(iv)(b)} only imposes finitely many conditions on
$\,\omega_{\psi,T}\,$. 

The definitions here do not require an $\mathcal{L}\mathcal{C}$-manifold, and
it is in principle quite permissible, albeit not usually possible, that
$\,\bigcup_{\psi\in J\setminus\Theta}U_0^\psi\,$ itself should be the whole
of $M$.

If \ref{extfam}\emph{(ii)} were weakened to require only that
$\,\{\theta\in\Theta:U^\psi\cap U_0^\theta\neq\emptyset\}\,$ be finite, with
a corresponding change in \ref{extsys}\emph{(iv)(b)}, the later arguments
could with some effort be improved to show that \emph{any} closed set in a
Finsler manifold is an $\mathcal{L}\mathcal{C}$-extension core in the sense
of \ref{core}. That conclusion is false, as I noted in \S 1; the induction in
\ref{prop} would indeed fail. But it is curious that the form of \emph{(ii)}
is so crucial for the extension theorems. \ 
\end{remark}

\begin{definition}
\ Again, let $M$ be a $\mathcal{D}$-manifold \label{strong}and $\mathfrak{V}$
an open cover of $M$. A \emph{strong} $\mathfrak{V}$-\emph{extension system}
in $M$ is a pair $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ satisfying
\ref{extfam}\emph{(i)}, \emph{(ii)}, \emph{(iii)} and\\
\hspace*{0.5cm}\emph{(iv}$'$\emph{)}\hspace*{0.5cm}for any $\;\psi\in
J\setminus\Theta\,$, there is a$\,$ $\mathcal{D}$-chart
$\;\omega_\psi:O_\psi\longrightarrow E\;$ for $M$ such that

\hspace*{5mm}\emph{(a)}\hspace*{0.5cm}there is some $\;V\in\mathfrak{V}\;$
such that $\;O_\psi\subseteq V\,$; and 

\hspace*{5mm}\emph{(b)}\hspace*{0.5cm}when $\;k,l\in J\;$ and $\;U^\psi\cap
U^k\neq\emptyset\neq U^k\cap U^l\,$, then $\;U^l\subseteq O_\psi\;$ and
$\;$co$(\omega_\psi(U_0^l))\prec\omega_\psi(U^l)\,$.
\end{definition}

\begin{proposition}
\ Su\label{prop}ppose $\mathcal{C}$ is an admissible category (see\emph{
\ref{admiss}}). Suppose that $\;\pi:L\longrightarrow M\;$ is a locally
trivial $\mathcal{L}\mathcal{C}$-fibration of
$\mathcal{L}\mathcal{C}$-manifolds, with base $M$ and fibre $N$ (see\emph{
\ref{product}}), where $N$ is $\mathcal{L}\mathcal{C}$-diffeomorphic to the
open convex subset $Y$ of the topological vector space $F$ and $M$ has the
locally convex Lindel\"of model $E$. 

Let $\mathfrak{V}$ be any cover of $M$ by open sets $V$ in $M$ for which
there is an $\mathcal{L}\mathcal{C}$-local trivialization
$\;\phi:\pi^{-1}(V)\longrightarrow V\times N\;$ of $\pi$. Suppose that the
pair $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ is a $\mathfrak{V}$-extension
system in $M$. Then, for any $\mathcal{L}\mathcal{C}$-section $\tau$ of $\pi$
over $\,U(\Theta)\,$, there exists an $\mathcal{L}\mathcal{C}$-section
$\sigma$ of $\pi$ over $M$ such that $\;\sigma\mid U_0(\Theta)=\tau\mid
U_0(\Theta)\,$.
\end{proposition}

\begin{proof}
\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{proem})}%
\hspace*{0.5cm}Taking $\;T:=(J\cap[1,\psi])\cup\left(\bigcup_{j\in
J\cap[1,\psi]}\Theta_j\right)\;$ in \ref{extsys}\emph{(iv)}, one finds that,
for the given $\mathfrak{V}$ and for each $\;\psi\in J\setminus\Theta\,$,
there exist both an $\mathcal{L}\mathcal{C}$-chart
$\;\omega_\psi:O_\psi\longrightarrow E\;$ (I write $\,\omega_\psi\,$ for
$\,\omega_{\psi,T}\,$ and $\,O_\psi\,$ for $\,O_{\psi,T}\,$) with the stated
properties, and an $\mathcal{L}\mathcal{C}$-local trivialization
$\;\phi_\psi:\pi^{-1}(O_\psi)\longrightarrow O_\psi\times Y\,$, where the
$\mathcal{L}\mathcal{C}$-diffeomorphism of the fibre with $Y$ has been
assimilated into $\,\phi_\psi\,$.\\
\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{heada})}%
\hspace*{0.5cm}Define, for $\;r=1,2,3,\ldots\,$, \ 
\begin{equation}\label{deft}
\left\{
\begin{aligned}
&T_r:=\{i:(1\leq i\leq r)\quad\text{or}\;\\
&\qquad\qquad(i\in\Theta\;\;\text{\&}\;\;(\exists\psi\notin\Theta)(1\leq\psi%
\leq r\;\:\text{\&}\:\;U^\psi\cap\,U^i\neq\emptyset\,))\}\,.
\end{aligned}
\right.\quad
\end{equation}
Clearly $\;\{1,2,3,\ldots,r\}\subseteq T_r\subseteq T_{r+1}\;$ for each
$\;r\geq 1\,$. Furthermore, hypothesis \ref{extfam}\emph{(ii)} implies that
$\,T_r\,$ must be finite. Let $\;\mathcal{A}(r):=\{U^i:i\in T_r\}\;$ and
$\;\mathcal{B}(r):=\{U_0^i:i\in T_r\}\,$, and recall the terminology of
\cite{atkin2}2.10. The inductive hypotheses at stage $r$ are as follows.\\
\ 

\hspace*{0.5cm}$\mathcal{H}_1$($r$):\hspace*{0.5cm}for every simplex $\Delta$
of $\,\mathcal{A}(r)\,$, there is an $\mathcal{L}\mathcal{C}$-section
$\,\sigma_r(\Delta)\,$ of $\pi$ over $\,A(r)^\Delta\,$, the star
$\,\bigcup_{i\in\Delta}U^i\,$ of the simplex $\Delta$ in $\,\mathcal{A}(r)\,$.

\hspace*{0.5cm}$\mathcal{H}_2$($r$):\hspace*{0.5cm}whenever
$\,\Delta_1,\Delta_2\,$ are simplices of $\,\mathcal{A}(r)\,$ and
$\;i\in\Delta_1\cap\Delta_2\,$, 
\begin{equation*}
\sigma_r(\Delta_1)\mid U_0^i=\sigma_r(\Delta_2)\mid U_0^i\,.
\end{equation*}
(The $\mathcal{L}\mathcal{C}$-sections $\,\sigma_r(\Delta)\,$ will therefore
fit together to define an$\;\mathcal{L}\mathcal{C}$-section $\,\sigma_r\,$
over the carrier $\;B(r):=\bigcup_{i\in T_r}U_0^i\,$).

\hspace*{0.5cm}$\mathcal{H}_3$($r$):\hspace*{0.5cm}for each simplex $\Delta$
of $\,\mathcal{A}(r)\,$, and any $\;i\in\Delta\cap\Theta\,$, 
\begin{equation*}
\sigma_r(\Delta)\mid U_0^i=\tau\mid U_0^i\,.
\end{equation*}
(In other words, $\;\;\sigma_r\mid U_0^i=\tau\mid U_0^i\;$ \ for each $\;i\in
T_r\cap\Theta\,$).\label{repairfrohrepsiforj}

\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{headb})}%
\hspace*{0.5cm}The induction begins with $\;r=0\,$, $\;T_0=\emptyset\,$,
$\;\mathcal{A}(0)=\mathcal{B}(0)=\emptyset\,$.\\
\ \ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{headc})}%
\hspace*{0.5cm}Suppose the inductive hypotheses satisfied for some $\;r\geq
0\,$. If $\;r+1\in T_r\,$, then $\;r+1\in\Theta\;$ and $\;T_{r+1}=T_r\,$. If
$\;r+1\notin T_r\,$, then $\;T_{r+1}\setminus T_r\;$ consists of $\,r+1\,$
and, if $\;r+1\notin\Theta\,$, \ also of those $\;\theta\in\Theta\;$ for
which $\,U^\theta\,$ meets $\,U^{r+1}\,$ but no earlier $\,U^\psi\,$ with
$\;\psi\notin\Theta\;$ (that is,
$\;\theta\in\Theta_{r+1}\setminus\left(\bigcup_{r\geq\psi\notin\Theta}\Theta_%
\psi\right)\,$). Thus the simplices of $\,\mathcal{A}(r+1)\,$ fall, by
(\ref{deft}), into three disjoint classes, which will be treated separately: 
\begin{equation*}
\begin{aligned}[t]
&\text{I.}&\quad&\text{simplices of }\,\mathcal{A}(r)\,\text{;}\\
&\text{II.}&\quad&\text{simplices not in }\,\mathcal{A}(r)\,\text{ of which
}\,r+1\,\text{ is not a vertex;}\\
&\text{III.}&\quad&\text{simplices not in }\,\mathcal{A}(r)\,\text{with
}\,r+1\,\text{ as a vertex.}
\end{aligned}
\end{equation*}
(Simplices of classes II and III only exist when $\;r+1\notin T_r\,$). \ \\
\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{headee})}%
\hspace*{0.5cm}If $\Delta$ is a simplex of $\,\mathcal{A}(r+1)\,$ in class I,
define $\;\sigma_{r+1}(\Delta):=\sigma_r$($\Delta)\,$.\\
\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{heade})}%
\hspace*{0.5cm}Next, suppose that $\Delta$ is a simplex of
$\,\mathcal{A}(r+1)\,$ in class II. As $\Delta$ is not a simplex of
$\,\mathcal{A}(r)\,$, it$\,$has at least one vertex $\;k>r+1\,$, necessarily
in $\Theta$, such that $\,U^k\,$ does not meet any $\,U^\psi\,$ for which
$\;1\leq\psi\leq r\;$ and $\;\psi\notin\Theta\,$. But this implies that such
vertices $\psi$ cannot belong to $\Delta$ at all, so that
$\;\Delta\subseteq\Theta\,$. Hence, $\;A(r+1)^\Delta\subseteq U(\Theta)\,$,
and one may define $\;\sigma_{r+1}(\Delta):=\tau\mid A(r+1)^\Delta\,$.\\
\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{headff}\label{headf})}%
\hspace*{0.5cm}The inductive hypotheses$\,\,\mathcal{H}_1$($r+1$)$\,$,
$\,\mathcal{H}_2$($r+1$)$\,$, and $\,\mathcal{H}_3$($r+1$)$\,$ are satisfied
when $\;\Delta_1,\Delta_2,\Delta\;$ are simplices in class I, because, by
\emph{(e)}, they reduce to $\,\mathcal{H}_1$($r$)$\,$,
$\,\mathcal{H}_2$($r$)$\,$, and $\,\mathcal{H}_3$($r$)$\,$. When $\Delta$ is
in class II, the definition \emph{(f)} instantly gives
$\,\mathcal{H}_3$($r+1$)$\,$. Ergo, $\,\mathcal{H}_2$($r+1$)$\,$ follows from
$\,\mathcal{H}_3$($r+1$)$\,$ and $\,\mathcal{H}_3$($r$)$\,$ when
$\,\Delta_1\,$ is in class II and $\,\Delta_2\,$ is in class I or class II,
since the vertices of $\,\Delta_1\,$ are all in $\Theta$ anyway. \\
\ 

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead}\label{headh})\label{headi}}%
\hspace*{0.5cm}The non-trivial case is that of a simplex of class III. Recall
that, for such a simplex to exist, $\;r+1\notin T_r\,$, and any vertex
$\;j>r+1\;$ must belong to $\,\Theta_{r+1}\,$. Let 
\begin{equation*}
H_r:=\{k\in T_{r+1}:(\exists j\in T_{r+1})(U^{r+1}\cap
U^j\neq\emptyset\;\;\text{\&}\;\;U^k\cap U^j\neq\emptyset\,)\}\,.
\end{equation*}
Take the $\mathcal{L}\mathcal{C}$-chart
$\;\omega_{r+1}:O_{r+1}\longrightarrow E\;$ and the
$\mathcal{L}\mathcal{C}$-local trivialization of $\pi$ over $\,O_{r+1}\,$,
$\;\phi_{r+1}:\pi^{-1}(O_{r+1})\longrightarrow O_{r+1}\times Y\,$, as before.
Thus, for every $\;k\in H_r\,$, 
\begin{equation}\label{wzorstar}
U^k\subseteq
O_{r+1}\;\quad\text{and}\quad\;\text{co}(\omega_{r+1}(U_0^k))\prec%
\omega_{r+1}(U^k)\,.
\end{equation}
Choose a convex open neighbourhood $W$ of $0$ in $E$ such that, for all
$\;k\in H_r\,$, 
\begin{equation}\label{stroninc}
R_{k,r}:=\text{co}(\omega_{r+1}(U_0^k))+W\prec\omega_{r+1}(U^k)\,.
\end{equation}
\ $\,R_{k,r}\,$ is a convex open set in $E$, and
$\;S_{k,r}:=\omega_{r+1}(U_0^k)\prec R_{k,r}\,$. Let
$\;P_{k,r}:=\omega_{r+1}(U^k)\,$.

Define complexes 
\begin{equation*}
\begin{aligned}[t]
&\mathcal{R}_r:=\{R_{k,r}:r+1\neq k\in H_r\}\,\text{,}\\
&\mathcal{S}_r:=\{S_{k,r}:r+1\neq k\in H_r\}\,\text{,}\\
&\mathcal{P}_r:=\{P_{k,r}:r+1\neq k\in H_r\}\,.
\end{aligned}
\end{equation*}
By (\ref{stroninc}), $\;\mathcal{S}_r\prec\mathcal{R}_r\,$, and
$\,\mathcal{R}_r\,$ consists of convex open sets. 

Take any simplex $\Delta$ of $\,\mathcal{P}_r\,$. Since $\;\omega_{r+1}\;$ is
injective, $\Delta$ is also a simplex of $\;\{U^k:r+1\neq k\in H_r\}\,$;
consequently, $\Delta$ is a simplex of $\,\mathcal{A}(r+1)\,$, and, as
$\;r+1\notin\Delta\,$, it must be a simplex of class I or class II. Hence
$\,\sigma_{r+1}(\Delta)\,$ has already been defined over $\,A(r+1)^\Delta\,$
at \emph{(e)} or \emph{(f)}. By \emph{(g)}, the
$\mathcal{L}\mathcal{C}$-sections $\,\sigma_{r+1}(\Delta)\,$ for simplices
$\Delta$ of$\,\,\mathcal{P}_r\,$ agree on the sets $\,U_0^k\,$ for $\;r+1\neq
k\in H_r\;$ (which is $\,\mathcal{H}_2$($r+1$)$\,$ for simplices of classes I
and II), and coincide with $\tau$ on $\,U_0^k\,$ for $\;r+1\neq k\in
H_r\cap\Theta\;$ (which is $\,\mathcal{H}_3$($r+1$)$\,$).

The $\mathcal{L}\mathcal{C}$-sections $\,\sigma_{r+1}(\Delta)\,$ may be
translated into $\mathcal{L}\mathcal{C}$-mappings
$\;\mu_{r+1}(\Delta):P_r^\Delta\longrightarrow Y\;$ ($\,P_r^\Delta\,$ is the
star of $\Delta$ in the complex $\,\mathcal{P}_r\,$), by the formula 
\begin{equation}\label{mudef}
\mu_{r+1}(\Delta):=p_2\circ\phi_{r+1}\circ\sigma_{r+1}(\Delta)\circ%
\omega_{r+1}^{-1}\,.
\end{equation}
The mappings $\,\mu_{r+1}(\Delta)\,$ fit together over the sets
$\,S_{k,r}\,$, and also agree with
$\,p_2\circ\phi_{r+1}\circ\tau\circ\omega_{r+1}^{-1}\,$ on $\,S_{k,r}\,$ when
$\;r+1\neq k\in H_r\cap\Theta\,$. Any simplex of$\,\,\mathcal{R}_r\,$ is
\emph{a fortiori} a simplex of $\,\mathcal{P}_r\,$, and $\,\mathcal{R}_r\,$
consists of open convex sets (although $\,\mathcal{P}_r\,$ need not), so
\cite{atkin2}5.1 applies with $\,\mathcal{R}_r\,$ in place of
``$\mathcal{A}$'', $\,\mathcal{S}_r\,$ instead of ``$\mathcal{B}$'', the
whole of $E$ as ``$X$'', and $\,\mu_{r+1}(\Delta)\,$ as
$\;$``$f_\Delta$''$\,$; hence, there is an $\mathcal{L}\mathcal{C}$-mapping
$\;h_r:E\longrightarrow Y\;$ which agrees with $\,\mu_{r+1}(\Delta)\,$ on
$\;S_{k,r}\,$, for any simplex $\Delta$ of$\,\,\mathcal{R}_r\,$ and any
vertex $k$ of $\Delta$. One may therefore use$\,\,h_r\,$ to define
$\,\sigma_{r+1}(\Gamma)\,$ \emph{simultaneously }for all simplices
$\Gamma\,$of $\,\mathcal{A}(r+1)\,$ of class III; for each such simplex
$\Gamma$, $\;A(r+1)^\Gamma\subseteq O_{r+1}\;$ by$\,$\ref{extsys}\emph{(iv)},
and one may set, for any $\;x\in A(r+1)^\Gamma\,$, 
\begin{equation}\label{sigmadef}
\sigma_{r+1}(\Gamma)(x):=\phi_{r+1}^{-1}(x,h_r\circ\omega_{r+1}(x))\,.
\end{equation}
This gives an $\mathcal{L}\mathcal{C}$-section of $\pi$ over
$\,A(r+1)^\Gamma\,$, and is the restriction of the
$\mathcal{L}\mathcal{C}$-section over $\,O_{r+1}\,$ defined by the formula on
the right-hand side. \\
$\,$

\emph{(\addtocounter{SEQhead}{1}\addtocounter{SEQhead}{1}\alph{SEQhead}%
\label{headj})}\hspace*{0.5cm}Let $\,\Gamma,\Delta\,$ be simplices of
$\,\mathcal{A}(r+1)\,$, and $\;j\in\Gamma\cap\Delta\,$. If both $\Gamma$ and
$\Delta$ are of class III, then (\ref{mudef}$)$ shows that
$\,\sigma_{r+1}(\Gamma)\,$ and $\,\sigma_{r+1}(\Delta)\,$ agree on $\,U^j\,$
and not just on $\,U_0^j\,$. 

If only $\Gamma$ is of class III, then $\;\Gamma\ni r+1\notin
T_r\cap\Delta\,$. Given any $\;i\in\Delta\,$, $\;U^i\cap U^j\neq\emptyset\,$,
and, as $\;j\in\Gamma\,$, $\;U^j\cap U^{r+1}\neq\emptyset\,$, so $\,i\in
H_r\;$ and $\;i\neq r+1\,$. Hence, by (\ref{wzorstar}),
$\;A(r+1)^\Delta\subseteq O_{r+1}\,$, and $\Delta$ is also a simplex of
$\,\mathcal{P}_r\,$ of class I or class II. However, $\,h_r\,$ was
constructed to agree with $\,\mu_{r+1}(\Delta)\,$ over each $\,S_{k,r}\,$ for
which $\;k\in\Delta\,$, and in particular over $\,S_{j,r}\,$; from the
formul\ae\ (\ref{mudef}) and (\ref{sigmadef}), this means that, for any
$\;x\in U_0^j\,$, $\;\omega_{r+1}(x)\in S_{j,r}\,$, 
\begin{equation*}
\begin{aligned}[t]
\sigma_{r+1}(\Gamma)(x):=\:&\phi_{r+1}^{-1}(x,h_r\circ\omega_{r+1}(x))\\
=\:&\phi_{r+1}^{-1}(x,p_2\circ\phi_{r+1}\circ\sigma_{r+1}(\Delta)(x))=%
\sigma_{r+1}(\Delta)(x)\,\text{,}
\end{aligned}
\end{equation*}
which, with \emph{(g)}, completes the proof of$\,\,\mathcal{H}_2$($r+1$)$\,$.
In the same way, $\,h_r\,$ was constructed to agree with
$\;p_2\circ\phi_{r+1}\circ\tau\circ\omega_{r+1}^{-1}\;$ when $\;r+1\neq k\in
H_r\cap\Theta\,$, so that 
\begin{equation*}
\sigma_{r+1}(\Gamma)\mid U_0^k=\tau\mid U_0^k\,.
\end{equation*}
This establishes the case of $\,\mathcal{H}_3$($r+1$)$\,$ not already settled
by \emph{(g)}. Consequently, the inductive hypotheses at stage $\,r+1\,$ have
been ensured by the construction at stage $r$, and it may be iterated to
produce a whole sequence of $\mathcal{L}\mathcal{C}$-mappings
$\,\sigma_r(\Delta)\,$.\\
$\,$

\emph{(\addtocounter{SEQhead}{1}\alph{SEQhead})}\hspace*{0.5cm}Recall the
definition of $\;\sigma_r\;$ at $\mathcal{H}_2$($r$). The inductive
construction above entails immediately that $\;\sigma_{r+1}\mid
U_0^i=\sigma_r\mid U_0^i\;$ for any $\;i\in T_r\,$. (Take $\;\Delta:=\{i\}\;$
in \emph{(d)}). Consequently, a section $\sigma$ of $\pi$ may be well-defined
by setting $\sigma$ to agree with $\,\sigma_r\,$ over $\,U_0^i\,$ when
$\;i\in T_r\,$. Since $\;\bigcup_{r=1}^{\,\infty}T_r=J\;$ by (\ref{deft}),
this defines $\sigma$ over $M$ (by \ref{extfam}\emph{(i)}); $\sigma$ is an
$\mathcal{L}\mathcal{C}$-mapping, since each $\,\sigma_r\,$ is
$\mathcal{L}\mathcal{C}$ by construction; and $\mathcal{H}_3$($r$) evidently
ensures that $\;\sigma\mid U_0(\Theta)=\tau\mid U_0(\Theta)\,$. 
\end{proof}

\begin{remark}
\label{rem2} T\label{remark}he induction above is reminiscent of
\cite{atkin2}4.2, because of the necessity of maintaining consistency of each
step of the construction with some fixed data (in this case $\tau$ is given,
in \cite{atkin2}4.3 $f$). A curiosity of the proof is that the local
extensions $\,h(\Delta)\,$ and $\,h_r\,$ do not need to be defined on the
whole of $E$, but only on $\,\omega_{r+1}(O_{r+1})\,$. However, this does not
permit any useful weakening of the hypotheses.

As here stated, \ref{prop} fails to mention a set $X$ such as appears in
\cite{atkin2}5.1. To incorporate it directly into a similar proposition, one
would have to make clumsy hypotheses on the coordinate transformations of
$M$. A relatively attractive consequence of that stronger version of
\cite{atkin2}5.1 is given later, \ref{badjob}. 

In the simple result \ref{modifiedprop} that follows, the topology on $F$ is
not really necessary --- it is included only for consistency with Definition
\ref{lcman}. With a little revision of the definitions, one could obtain the
Proposition for functions into any vector space. The proof of
\ref{modifiedprop} is exactly as in \ref{prop}, with all mention of local
trivializations removed. 

There are also versions of \ref{prop} corresponding to \cite{atkin2}5.5,
which I shall not discuss here because they require some further structure on
the manifold, and an extended version corresponding to \cite{atkin2}4.6,
mentioned after \ref{coreexten}. \ \ 
\end{remark}

\begin{proposition}
\ L\label{modifiedprop}et $\mathcal{C}$ be an admissible category, and let
$M$ be an $\mathcal{L}\mathcal{C}$-manifold with locally convex Lindel\"of
model $E$. Let $Y$ be a convex set in the topological vector space $F$.
Suppose $\,(\{U^i\},\{U_0^i\},\Theta)\,$ is an extension system in $M$. Then,
for any $\mathcal{L}\mathcal{C}$-mapping $\;g:U(\Theta)\longrightarrow F\;$
with image in $Y$, there exists an $\mathcal{L}\mathcal{C}$-mapping
$\;f:M\longrightarrow F\;$ with image in $Y$ such that $\;f\mid
U_0(\Theta)=g\mid U_0(\Theta)\,$. \hfill{\qedsymbol}
\end{proposition}

\begin{definition}
\label{guess}\label{core}\label{ec} L\label{gg}et $M$ be an
$\mathcal{L}\mathcal{C}$-manifold. Say that a set $X$ in $M$ is an
$\mathcal{L}\mathcal{C}$-\emph{extension core} in $M$ (for
$\mathcal{L}\mathcal{C}$-sections of locally trivial
$\mathcal{L}\mathcal{C}$-fibrations with fibres
$\mathcal{L}\mathcal{C}$-diffeomorphic to open convex sets in topological
vector spaces) if it is closed and, for any open neighbourhood $U$ of $X$ in
$M$, there is an open neighbourhood $V$ of $X$ such that, for any
$\mathcal{L}\mathcal{C}$-section $\tau$ over $U$ of a locally trivial
$\mathcal{L}\mathcal{C}$-fibration $\;\pi:L\longrightarrow M\;$ with fibre
$\mathcal{L}\mathcal{C}$-diffeomorphic to an open convex subset $Y$ of a
locally convex space $F$, there is an $\mathcal{L}\mathcal{C}$-section
$\sigma$ of $\pi$ over $M$ for which $\;\sigma\mid V=\tau\mid V\,$.
\end{definition}

This definition may be varied by considering different kinds of $\tau$ and
$\sigma$, such as real-valued $\mathcal{L}\mathcal{C}$-functions or sections
of locally trivial $\mathcal{L}\mathcal{C}$-fibrations with more general
fibres. In the rest of the paper, I shall consider a condition which is
sufficient to ensure that a set is an $\mathcal{L}\mathcal{C}$-extension core
either in the sense above, or for $\mathcal{L}\mathcal{C}$-functions with
values in a topological vector space. \ \ \ 

\begin{definition}
\ Th\label{extensible}e subset $X$ of the $\mathcal{D}$-manifold $M$ is
\emph{simply }$\mathcal{D}$\emph{-extensible}, where $\mathcal{D}$ is an
applicable category, if $X$ is closed in $M$ and, for any open cover
$\mathfrak{V}$ of $M$ and open neighbourhood $V$ of $X$, there is a strong
$\mathfrak{V}$-extension system $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ (see
\ref{strong}) such that $\;X\subseteq U_0(\Theta)\;$ and
$\;U(\Theta)\subseteq V\,$.
\end{definition}

\begin{remark}
\ T\label{definition}his condition involves only $X$ and $M$, and not a
specific extension problem; to prove that $X$ is an
$\mathcal{L}\mathcal{C}$-extension core in a particular sense, it is commonly
not necessary to demand \emph{strong }extension systems or to allow
\emph{arbitrary} open covers $\mathfrak{V}$. The proof of \ref{prop} only
requires the members of $\mathfrak{V}$ to admit
$\mathcal{L}\mathcal{C}$-local trivializations, which in some circumstances
will exist over certain sets for any locally trivial fibration, for instance
because of contractibility. In short, simple
$\mathcal{L}\mathcal{C}$-extensibility is often an excessively strong
condition for $\mathcal{L}\mathcal{C}$-extension cores of a given kind, but
it works rather generally.

If $M$ is, as usually, T$_4$, explicit mention of $V$ in \ref{extensible} may
seem superfluous, as one could consider the finer cover $\,\mathfrak{V}'\,$
consisting of all the sets $\,V\cap W\,$ and $\,W\setminus M\,$ as $W$ varies
over $\mathfrak{V}$. However, it might then be necessary to diminish $\Theta$
in order to ensure that $\;U(\Theta$)$\subseteq V\,$, and the removal of some
element $\theta$ from $\Theta$ may invalidate \ref{extfam}\emph{(ii)}, since
there is no reason why $\,U^\theta\,$ should meet only finitely many other
sets $\,U^\phi\,$ for which $\;\phi\in\Theta\,$. If one were to add to
\ref{strong} a requirement that $\;\Theta:=\{j\in J:U^j\cap
X\neq\emptyset\}\,$, this objection ceases to apply and mention of $V$ in
\ref{extensible} could be suppressed. The proposed extra requirement is
fairly natural and is true in the specific constructions of \S 5, but is not
otherwise necessary. 
\end{remark}

\begin{proposition}
\ S\label{coreexten}uppose $\mathcal{C}$ is an admissible category (see\emph{
\ref{admiss}}), and $M$ is an $\mathcal{L}\mathcal{C}$-manifold with locally
convex Lindel\"of model. If $X$ is a simply
$\mathcal{L}\mathcal{C}$-extensible subset of $M$, then $X$ is an
$\mathcal{L}\mathcal{C}$-extension core in $M$.

Furthermore, for any $\mathcal{L}\mathcal{C}$-map $\,g:U\longrightarrow F\,$,
where $F$ is a topological vector space and $U$ is a neighbourhood of $X$ in
$M$, there exists an $\mathcal{L}\mathcal{C}$-map $\;f:M\longrightarrow F\;$
such that $\;f\mid V=g\mid V\;$ (where $\;V\subseteq U\;$ is the
neighbourhood of $X$ which is specified in the definition\emph{ \ref{ec}})
and $\;f(M)\subseteq\,$\emph{co}$(g(U))\,$.\hfill{\qedsymbol}
\end{proposition}

\label{normal}The section $\sigma$ or the mapping $f$ is obtained from $\tau$
or $g$ by a procedure of taking convex combinations of local extensions (see
\ref{prop} and \cite{atkin2}, \emph{passim}) in a way which does not depend
on $\tau$ or $g$, but only on $\,X,U\,$ and $M$; hence, if the local
extensions depend affine-linearly on $\tau$ or $g$, so do the global
extensions. In this paper, the existence of linear local extension operators
in the sense of \cite{atkin2}4.4 is assured for
$\mathcal{L}\mathcal{C}$-sections or $\mathcal{L}\mathcal{C}$-functions by
linear algebra, since there are surjective linear restriction operators by
definition.

My purpose now is to establish criteria for sets to be simply
$\mathcal{L}\mathcal{C}$-extensible. \label{not}I shall concentrate my
attention on$\,$manifolds which are modelled on normed spaces and C$^1$ in
the sense of Fr\a'echet calculus; for them the condition
\ref{strong}\emph{(iv}$'$\emph{)(b) }is rather easily satisfied, because a
C$^1$ mapping deforms small norm-balls whilst approximately preserving
convexity. This property is most easily described by Finsler structures.
(There are other less natural categories $\mathcal{C}$ for which similar but
more complicated procedures are possible).

\ 

\section{$\boldsymbol{\epsilon}$-flatness in Finsler manifolds.}

\begin{definition}
\ L\label{strongchart}et $M$ be a C$^0$ manifold modelled on a topological
space $Y$. A chart for $M$ is then just a homeomorphism of an open set in $M$
with an open set in $Y$. Define a chart to be \emph{strong in} $M$ if it is a
chart $\;\psi:U\longrightarrow Y\;$ (where $U$ is open in $M$ and
$\,\psi(U)\,$ open in $Y$) such that, whenever $C$ is a closed set in $M$ and
$\;C\subseteq U\,$, then $\,\psi(C)\,$ is closed in $Y$; and, conversely,
whenever $D$ is a closed set in $Y$ which is included in $\,\psi(U)\,$, then
$\,\psi^{-1}(D)\,$ is closed in $M$. \ 
\end{definition}

This definition is useful because charts may ``misbehave near the boundary''.
In finite dimensions, the problem is customarily avoided by making use of
compact rather than closed sets.

A C$^0$ atlas $\mathfrak{A}$ for $M$ is \emph{hereditary }if, whenever
$\;\psi:U\longrightarrow\psi(U)\subseteq Y\;$ is a chart belonging to
$\mathfrak{A}$ and $V$ is an open subset of $U$, $\;\psi\mid
V:V\longrightarrow\psi(V)\;$ is also in $\mathfrak{A}$. 

\begin{lemma}
\ S\label{strongsubchart}uppose the spaces $X$ and $Y$ are both T$_4$. Let
$U$ be open in $X$ and $V$ in $Y$, and let $\;\psi:U\longrightarrow V\;$ be a
homeomorphism. If $C$ is a closed subset of $X$ included in $U$ and such that
$\,\psi(C)\,$ is closed in $Y$, then there exists an open neighbourhood $W$
of $C$ in $U$ such that $\psi$ establishes a one-one correspondence between
the closed sets of $X$ included in $W$ and the closed sets of $Y$ included in
$\,\psi(W)\,$. Similarly, if $M$ is a regular \emph{C}$^0$\emph{ }manifold,
$M$ an open submanifold, and $\mathfrak{A}$ a hereditary atlas for $M$, the
charts of $\mathfrak{A}$ which are strong in $M$ also form an atlas for $M$.
Finally, the restriction of a chart strong in $M$ to an open subset of its
domain is also a chart strong in $M$.
\end{lemma}

\begin{proof}
As $M$ and $X$ are T$_4$, take neighbourhoods $\,W_1\,$ of $C$ in $X$, and
$\,W_2\,$ of $\,\psi(C)\,$ in $Y$, such that the closure $\,\overline{W_1}\,$
of $\,W_1\,$ in $X$ is within $U$ and the closure $\,\overline{W_2}\,$ of
$\,W_2\,$ in $Y$ is within $\,\psi(U)\,$. Take
$\;W:=W_1\cap\psi^{-1}(W_2)\,$. It is easy to check that $W$ has the required
property. For the second assertion, it suffices to replace ``$C$'' by an
arbitrary point ``$x$'', and the argument then only requires regularity. 
\end{proof}

\begin{definition}
\label{adequate} T\label{repair3}he category $\mathcal{D}$ is \emph{adequate}
if it is applicable (see \ref{applic}), its objects are open subsets in
normed spaces, and all its morphisms are C$^1$ in the Fr\a'echet sense.
\end{definition}

For the rest of this section $M$ is to be a paracompact, but not necessarily
separable, $\mathcal{D}$-manifold, where $\mathcal{D}$ is an adequate
category and $M$ is modelled on the normed space $E$. ($M$ and $E$ are
therefore regular). There is no need here for $\mathcal{D}$ to have a local
extensibility property. $M$ admits a tangential Finsler structure $p$ (for
the infinite-dimensional case, see, for instance, \cite{palais} or
\cite{atkin1}), which induces a corresponding Finsler metric $d$. The metric
ball of radius $r$ about $\;x\in M\;$ will be denoted by $\,B(x;r)\,$. (In
\ref{prop} a similar notation $\,B(r)\,$ was used with a quite different
meaning). I shall write $\,p_x(\xi)\,\,$or $\,p(x,\xi)\,$ to denote the
Finsler-norm of the vector $\xi$ in the tangent space $\,T_xM\,$ to $M$ at
$x$.

\begin{definition}
\ S\label{flat}uppose that $\;\epsilon\geq 0\,$. A $\mathcal{D}$-chart
$\;\psi:U\longrightarrow E\;$ of $M$ is $\epsilon$\emph{-flat }if it is
strong in $M$ and, for any two points $\;y,z\in U\,$, and any $\;\xi\in E\,$, 
\begin{equation}\label{wzorflat}
p_y((T_y\psi)^{-1}\xi)\leq(1+\epsilon)p_z((T_z\psi)^{-1}\xi)\,.
\end{equation}
\end{definition}

\begin{lemma}
\ A\label{flatbase}ny subchart of an $\epsilon$-flat $\mathcal{D}$-chart of
$M$ is itself $\epsilon$-flat. If $\;\epsilon>0\,$, $\;\psi:U\longrightarrow
E\;$ is a $\mathcal{D}$-chart of $M$, and $\;x\in U\,$, then $x$ has a base
of open neighbourhoods in $U$ on each of which the restriction of $\psi$ is
$\epsilon$-flat.\label{ggg}
\end{lemma}

\begin{proof}
Any restriction to an open subset of a chart in $\mathfrak{A}$ satisfying
(\ref{wzorflat}) and strong in $M$ is also a chart strong in $M$, by
\ref{strongsubchart}, and evidently still satisfies (\ref{wzorflat}). It
therefore suffices to prove the existence of one $\epsilon$-flat
neighbourhood of $x$. Take any strong chart $\;\psi:U\longrightarrow E\,$,
where $\;x\in U\,$, and then, using the definition of a tangential Finsler
structure (\cite{palais} or \cite{atkin1}), let $V$ be an open neighbourhood
of $x$ in $U$ such that, for all $\;y\in V\;$ and all $\;\xi\in E\,$, 
\begin{equation*}
(1+\epsilon)^{-1/2}p_y(\text{(}T_y\psi)^{-1}\xi)\leq
p_x((T_x\psi)^{-1}\xi)\leq(1+\epsilon)^{1/2}p_y((T_y\psi)^{-1}\xi)\,.
\end{equation*}
Then $\;\psi\mid V\;$ is an $\epsilon$-flat chart.
\end{proof}

\begin{lemma}
\ L\label{omisery}et $\;\omega:B(x;R)\longrightarrow E\;$ be a
$\mathcal{D}$-chart of $M$. Norm $E$ by 
\begin{equation}\label{normdef}
\left\|\xi\right\|_{\omega,x}:=p_x((T_x\omega)^{-1}\xi)\,\text{,}
\end{equation}
and let $\,B_{\omega,x}'(\eta;\kappa)\,$ denote the ball of radius $\kappa$
about $\;\eta\in E\;$ with respect to $\,\left\|\xi\right\|_{\omega,x}\,$. If
$\omega$ is $\epsilon$-flat, then, for any $\;r\in[0,R)\,$, 
\begin{equation}\label{incl}
B_{\omega,x}'(\omega(x);(1+\epsilon)^{-1}r)\subseteq\omega(B(x;r))\subseteq\,%
B_{\omega,x}'(\omega(x);(1+\epsilon)r)\,.
\end{equation}
\end{lemma}

\begin{proof}
(\ref{wzorflat}) ensures that, for each $\;y\in B(x;r)\,$, and each
$\;\eta\in T_yM\,$, 
\begin{equation}\label{flatness}
(1+\epsilon)^{-1}\left\|T_y\omega\cdot\eta\right\|_{\omega,x}\leq
p_y(\eta)\leq(1+\epsilon)\left\|T_y\omega\cdot\eta\right\|_{\omega,x}\,.
\end{equation}
(Take $\;\xi:=T_y\omega\cdot\eta\;$ to see this). If $\;y\in B(x;R)\;$ and
$\;\delta>0\,$, and $\,\gamma(t)\,$, for$\;0\leq t\leq 1\,$, is a piecewise
C$^1$ path in $\;B(x;R)\,$ from $x$ to $y$ of $p$-length less than
$\;d(x,y)+\delta\,$, then, from the first inequality of (\ref{flatness}),
$\,\omega\circ\gamma\,$ has $\left\|\,\right\|_{\omega,x}$-length not
exceeding $\,(1+\epsilon)(d(x,y)+\delta)\,$, and \ 
\begin{equation*}
\left\|\omega(x)-\omega(y)\right\|_{\omega,x}\leq(1+\epsilon)(d(x,y)+\delta)\,%
.
\end{equation*}
Since $\delta$ may be arbitrarily small, the second inclusion of (\ref{incl})
follows.

Let $\;a\in E\,$, $\;\left\|a\right\|_{\omega,x}=1\,$. Define the ray
$\,\gamma_a\,$ by$\,\;\gamma_a(t):=\omega(x)+ta\,$, for $\;t\geq 0\,$. Let \ 
\begin{equation}\label{tausuba}
\begin{aligned}[t]
\tau_a&:=\sup\{t\geq 0:\gamma_a([0,t])\subseteq\omega(B(x;r))\}\,.
\end{aligned}
\end{equation}
Since $\,\omega(B(x;r))\,$ is open in $E$, certainly $\;\tau_a>0\,$. Consider
the lifted path 
\begin{equation*}
\widehat{\gamma}_a:=\omega^{-1}\circ\gamma_a:[0,\tau_a)\longrightarrow
B(x;r)\,.
\end{equation*}
This is a C$^1$ path without end-point. Its length in $M$ is 
\begin{equation*}
\int_0^{\tau_a}p(\widehat{\gamma}_a(t),(T_{\widehat{\gamma}_a(t)}%
\omega)^{-1}a)\,dt\leq\int_0^{\tau_a}(1+\epsilon)\left\|a\right\|_{\omega,x}\,%
dt=(1+\epsilon)\tau_a\,\text{,}
\end{equation*}
by the second inequality of (\ref{flatness}). Thus
$\;\,\widehat{\gamma}_a([0,\tau_a))\subseteq B(x;(1+\epsilon)\tau_a)\,$. 

Suppose, if possible, that $\;(1+\epsilon)\tau_a<r\,$. The closure $C$ of
$\,B(x;(1+\epsilon)\tau_a)\,$ in $M$ is then included in $\,B(x;r)\,$. As
$\omega$ is a chart strong in $M$ (by definition \ref{flat}; see also
\ref{strongchart}), $\,\omega(C)\,$ is closed in $E$. Since
$\;\widehat{\gamma}_a([0,\tau_a))\subseteq C\,$, certainly
$\;\gamma_a([0,\tau_a))\subseteq\omega(C)\,$. Hence
$\;\gamma_a(\tau_a)\in\omega(C)\subseteq\omega(B(x;r))\,$, which is open in
$E$, so $\;\gamma_a([0,t])\subseteq\omega(B(x;r))$ \ for some values of $t$
greater than $\;\tau_a\,$, contradicting (\ref{tausuba}). Therefore
$\;(1+\epsilon)\tau_a\geq r\,$. This means, by (\ref{tausuba}), that
$\;ta\in\omega(B(x;r))\;$ for $\;0\leq t<(1+\epsilon)^{-1}r\,$, and proves
the result.
\end{proof}

\begin{lemma}
\label{distances} Le\label{goto}t $\;\epsilon\in[0,\sqrt{2}-1]\,$, and
suppose that $\;\omega:B(x;2\delta)\longrightarrow E\;$ is an $\epsilon$-flat
$\mathcal{D}$-chart of $M$. Then, for any $\;y,z\in B(x;\delta)\,$, 
\begin{equation*}
(1+\epsilon)^{-1}\left\|\omega(y)-\omega(z)\right\|_{\omega,x}\leq
d(y,z)\leq(1+\epsilon)\left\|\omega(y)-\omega(z)\right\|_{\omega,x}\,.
\end{equation*}
\end{lemma}

\begin{proof}
Since $\;d(y,z)<2\delta\;$ by the triangle inequality, $\,d(y,z)\,$ is the
infimum of lengths of piecewise C$^1$ paths shorter than $\,2\delta\,$
between $y$ and $z$. Any such path must stay wholly within
$\,B(x;2\delta)\,$; otherwise it would join $y$ to a point $w$ outside
$\,B(x;2\delta)\,$, by a segment necessarily of length greater than $\delta$,
and then $w$ to $z$, by a second segment of length more than $\delta$; thus,
the whole path would have to be of length greater than $\,2\delta\,$.

Let $\;\gamma:[0,1]\longrightarrow M\;$ be a piecewise C$^1$ path in
$\;B(x;2\delta)\subseteq M\,$. Introduce a norm in $E$ by (\ref{normdef}).
The Finsler length of $\gamma$ with respect to $p$, $\,\mathbb{L}(\gamma)\,$,
is given by the formula 
\begin{equation*}
\mathbb{L}(\gamma):=\int_0^1p_{\gamma(t)}(\gamma'(t))\,dt\,\text{,}
\end{equation*}
and, by (\ref{flatness}), it follows that 
\begin{equation*}
\mathbb{L}(\gamma)\geq(1+\epsilon)^{-1}\int_0^1\left\|T_{\gamma(t)}\omega%
\cdot\gamma'(t)\right\|_{\omega,x}\,dt\,.
\end{equation*}
The right-hand integral is the $_{\omega,x}$-length in $E$ of
$\,\omega\circ\gamma\,$, which cannot be less than
$\;\left\|\omega(\gamma(1))-\omega(\gamma(0))\right\|_{\omega,x}\,$.
(Rectilinear segments in $E$ minimize the norm-length of C$^1$ paths between
their end-points --- from the triangle inequality and Riemann sums, for
instance). Thus, for any piecewise C$^1$ path $\gamma$ in $\;B(x;2\delta)\,$, 
\begin{equation*}
\mathbb{L}(\gamma)\geq(1+\epsilon)^{-1}\left\|\omega(\gamma(1))-\omega(%
\gamma(0))\right\|_{\omega,x}\,.
\end{equation*}
Since $\,d(y,z)\,$ is the infimum over such paths $\gamma$ between $z$ and
$y$, the first of the stated inequalities is established.

Now join $y$ to $x$ by a piecewise C$^1$ path $\;\beta:[0,1]\longrightarrow
M\;$ of Finsler-length less than $\delta$. Then \\
\begin{equation*}
\begin{aligned}[t]
\left\|\omega(y)-\omega(x)\right\|_{\omega,x}&\leq\int_0^1\left\|(\omega\circ%
\beta)'(t)\right\|_{\omega,x}\,dt=\int_0^1\left\|T_{\beta(t)}\omega\cdot%
\beta'(t)\right\|_{\omega,x}\,dt\\
&\leq(1+\epsilon)\int_0^1p_{\beta(t)}(\beta'(t))\,dt=(1+\epsilon)\mathbb{L}(%
\beta)<(1+\epsilon)\delta\,.
\end{aligned}
\end{equation*}
The same holds for $\;\left\|\omega(z)\;-\text{
}\omega(x)\right\|_{\omega,x}\,$. Hence, the straight-line segment $\theta$
between $\,\omega(y)\,$ and $\,\omega(z)\,$, defined by
$\;\theta(t):=t\omega(z)+(1-t)\omega(y)\;$ for $\;0\leq t\leq 1\,$, lies
entirely within the open ball of radius $\,(1+\epsilon)\delta\,$ about
$\,\omega(x)\,$ in $E$. By hypothesis, $\;(1+\epsilon)\delta\leq
2\delta/(1+\epsilon)\,$; hence, from \ref{omisery}, all points of the segment
lie in $\,\omega(B(x;2\delta))\,$, so that the path
$\;\zeta:=\omega^{-1}\circ\theta\;$ joining $y$ to $z$ is defined and C$^1$
in $\;B(x;2\delta)\;$. Apply (\ref{flatness}): 
\begin{equation*}
\begin{aligned}[t]
d(y,z)\leq\mathbb{L}(\zeta)=\int_0^1p_{\zeta(t)}(\zeta'(t))\,dt&\leq(1+%
\epsilon)\int_0^1\left\|T_{\zeta(t)}\omega\cdot\zeta'(t)\right\|_{\omega,x}\,%
dt\\
=(1+\epsilon)\int_0^1\left\|\theta'(t)\right\|_{\omega,x}\,dt&=(1+\epsilon)%
\left\|\omega(z)-\omega(y)\right\|_{\omega,x}\,\text{,}
\end{aligned}
\end{equation*}
and this is exactly the second inequality required.
\end{proof}

\begin{definition}
\ Given $\;t\in\mathbb{R}\;$ and $\;a\in E\,$, the \emph{affine expansion by
}$t$\emph{ about }$a$ is the mapping $\;A(t,a):E\longrightarrow E:\xi\mapsto
t\xi+(1-t)a\,$.
\end{definition}

\begin{remark}
\ If\label{rk} $C$ is a convex subset of $E$ and $\,a\in C$, then, whenever
$\;t>1\,$, $\;C\prec A(t,a)C\,$. 
\end{remark}

\begin{corollary}
\ L\label{ball}et $\,r,R,\delta,\epsilon\,$ be positive numbers such that
$\;\epsilon\leq\sqrt{2}-1\;$ and$\,$
\begin{equation*}
2(1+\epsilon)^3(r+\delta)\leq R\,.
\end{equation*}
Then, if $\;\omega:B(x;R)\longrightarrow E\;$ is an $\epsilon$-flat chart and
$\;y\in B(x;r)\,$, 
\begin{equation*}
A(1+\epsilon,\omega(y))\text{\emph{co}}(\omega(B(y;\delta))\subseteq%
\omega(B(y;(1+\epsilon)^3\delta)\,.
\end{equation*}
\end{corollary}

\begin{proof}
Certainly $\;B(y;\delta)\subseteq B(x;r+\delta)\,$, and $\;2(r+\delta)\leq
R\,$. By the first inequality of \ref{distances},
$\;\omega(B(y;\delta))\subseteq
B_{\omega,x}'(\omega(y);(1+\epsilon)\delta$)$\,$, and, therefore, 
\begin{equation}\label{convex}
\text{co}(\omega(B(y;\delta))\subseteq
B_{\omega,x}'(\omega(y);(1+\epsilon)\delta)\,.
\end{equation}
But, by \ref{omisery}, $\;\omega(y)\in
B_{\omega,x}'(\omega(x);(1+\epsilon)r)\,$, and so 
\begin{equation*}
\begin{aligned}[t]
B_{\omega,x}'(\omega(y);(1+\epsilon)^2\delta)&\subseteq
B_{\omega,x}'(\omega(x);(1+\epsilon)^2(r+\delta))\\
&\subseteq\omega(B(x;(1+\epsilon)^3(r+\delta)))\,.
\end{aligned}
\end{equation*}
So, if $\;\zeta\in B_{\omega,x}'(\omega(y);(1+\epsilon)^2\delta)\,$, there is
some $\;z\in B(x;(1+\epsilon)^3(r+\delta))\;$ such that
$\;\omega(z)=\zeta\,$, and, by the second inequality of \ref{distances},
$\;z\in B(y;(1+\epsilon)^3\delta)\,$. Hence 
\begin{equation}\label{hiccup}
B_{\omega,x}'(\omega(y);(1+\epsilon)^2\delta)\subseteq\omega(B(y;(1+%
\epsilon)^3\delta))\,.
\end{equation}
From (\ref{convex}), 
\begin{equation*}
\begin{aligned}[t]
A(1+\epsilon,\omega(y))\text{co}(B(\omega(y);\delta))&\subseteq
A(1+\epsilon,\omega(y))B_{\omega,x}'(\omega(y);(1+\epsilon)\delta)\\
&=B_{\omega,x}'(\omega(y);(1+\epsilon)^2\delta)\,\text{,}
\end{aligned}
\end{equation*}
which, with (\ref{hiccup}), proves the result.
\end{proof}

\begin{definition}
\ L\label{modflat}et $\mathfrak{A}$ be a hereditary $\mathcal{D}$-atlas for
$M$. The \emph{modulus of }$\epsilon$\emph{-flatness }of $M$ with respect to
$\mathfrak{A}$ and $p$ is the function
$\;\rho_\epsilon:M\longrightarrow\mathbb{R}\;$ defined as follows. For each
$\;x\in N\,$, consider the class of positive numbers $\delta$ such that
$\,B(x;\delta)\,$ is the domain of an $\epsilon$-flat chart of
$\mathfrak{A}$. This class has a supremum$\,\,\rho'_\epsilon(x)\,$, which may
be $\,\infty$ or $\,-\infty\,$. Set
$\;\rho_\epsilon(x):=\min(1,\rho'_\epsilon(x))\,$. 
\end{definition}

\begin{lemma}
\ I\label{lip}f $\;\epsilon>0\,$, the modulus of $\,\epsilon$-flatness
$\,\rho_\epsilon\,$ with respect to the Finsler structure $p$ and the atlas
$\mathfrak{A}$ takes strictly positive values, and satisfies the Lipschitz
condition that, for any $\;x,y\in N\,$,
$\;|\rho_\epsilon(x)-\rho_\epsilon(y)|\leq d(x,y)\,$.
\end{lemma}

\begin{proof}
If $\;\psi:U\longrightarrow E\;$ is a chart of $\mathfrak{A}$, where $\;x\in
U\,$, then by \ref{flatbase} there is an open neighbourhood $V$ of $x$ in $U$
such that $\,\psi\mid V\,$ is $\epsilon$-flat. Take $\;\delta>0\;$ such that
$\;B(x;\delta)\subseteq V\,$; then $\;\rho'_\epsilon(x)\geq\delta\,$. This
proves the first assertion.

Suppose $\;d(x,y)<\rho'_\epsilon(x)\,$; let
$\;\kappa\in(d(x,y),\rho'_\epsilon(x))$. By hypothesis, there is an
$\epsilon$-flat chart of $\mathfrak{A}$, $\;\phi:B(x;\kappa)\longrightarrow
E\,$. Now $\;B(y;\kappa-d(x,y))\subseteq B(x;\kappa)\,$; hence,$\,\,\phi\mid
B(x;\kappa-d(x,y))\,$ is also $\epsilon$-flat and in $\mathfrak{A}$, and
$\;\rho'_\epsilon$($y)\geq\kappa-d(x,y)\,$. Since $\kappa$ was arbitrary in
$\,(d(x,y),\rho'_\epsilon(x))\,$,
$\;\rho'_\epsilon(y)\geq\rho'_\epsilon(x)-d(x,y)\,$. This is also trivially
true if $\;d(x,y)\geq\rho'_\epsilon(x)\,$.

Consequently, either $\,\rho'_\epsilon(x)\,$ is infinite for all $x$, or it
is finite for all $x$. In the second case, by symmetry,
$\;|\rho'_\epsilon(x)-\rho'_\epsilon(y)|\leq d(x,y)\,$. The desired result
for $\,\rho_\epsilon(x)\,$ follows.\label{here}
\end{proof}

\begin{lemma}
\label{meetingball}Su\label{hamilton}ppose
$\;\sigma:M\longrightarrow[0,\infty)\;$ is a function satisfying the
Lipschitz condition that $\;|\sigma(x)-\sigma(y)|\leq d(x,y)\;$ for all
$\;x,y\in M\,$. Let $\,0\leq\alpha<1\,$. If $\;x_i\in M\;$ for
$\;i=0,1,2,3,4\,$, and, for $\;i=0,1,2,3\,$, 
\begin{equation*}
B(x_i;\alpha\sigma(x_i))\cap
B(x_{i+1};\alpha\sigma(x_{i+1}))\neq\emptyset\,\text{,}
\end{equation*}
\ then, for $\;0\leq i\leq 4\,$, 
\begin{equation*}
\textstyle\sigma(x_i)<\left(\displaystyle\frac{1+\alpha}{1-\alpha}\right)^i%
\sigma(x_0)\quad\text{and}\quad B(x_4;\alpha\sigma(x_4))\subseteq
B(x_0;\beta\sigma(x_0))\,\text{,}
\end{equation*}
where 
\begin{equation*}
\beta:=\alpha\left(\frac{9+4\alpha+14\alpha^2+4\alpha^3+\alpha^4}{(1-%
\alpha)^4}\right)\,.
\end{equation*}
\ \ 
\end{lemma}

\begin{proof}
Certainly $\;d(x_0,x_1)<\alpha\sigma(x_0)+\alpha\sigma(x_1)\,$, so that from
the Lipschitz condition 
\begin{equation*}
\sigma(x_1)<\sigma(x_0)+\alpha\sigma(x_0)+\alpha\sigma(x_1)\,\text{,}
\end{equation*}
and consequently $\;\sigma(x_1)<\frac{1+\alpha}{1-\alpha}\,\sigma(x_0)\,$,
and similarly $\;\sigma(x_2)<\frac{1+\alpha}{1-\alpha}\,\sigma(x_{_1})\;$ and
so on. Hence, if $\;y\in B(x_4;\alpha\sigma(x_4))\,$, 
\begin{equation*}
d(x_0,y)<\alpha\left(1+2\frac{1+\alpha}{1-\alpha}+2\left(\frac{1+\alpha}{1-%
\alpha}\right)^2+2\left(\frac{1+\alpha}{1-\alpha}\right)^3+2\left(\frac{1+%
\alpha}{1-\alpha}\right)^4\right)\sigma(x)\,\text{,}
\end{equation*}
which is, in effect, the result stated. 
\end{proof}

This result trivially implies similar statements for smaller numbers of balls.

\ 

\section{Simply $\boldsymbol{\mathcal{D}}$-extensible sets.}

\begin{proposition}
\ L\label{loccpctunion}et $M$ be a paracompact $\mathcal{D}$-manifold, where
$\mathcal{D}$ is an applicable category (see\emph{ \ref{applic}}). Let $X$ be
a closed subset of $M$, and let $K$ be a closed locally compact subset of
$M$. If $X$ is simply $\mathcal{D}$-extensible, then so is $\,X\cup K\,$.
\end{proposition}

A finite-dimensional submanifold, in the sense of \ref{subman}, of a
(topological) manifold $M$ modelled on a topological vector space (one takes
$\mathcal{D}$ in \ref{subman} to consist of all continuous maps of open sets)
is necessarily a closed locally compact set in $M$. (The corresponding
subspace of the model is closed, being finite-dimensional). 

\begin{proof}
\emph{(\addtocounter{SEQnorth}{1}\alph{SEQnorth}\label{seca})}%
\hspace*{0.5cm}Suppose that $V$ is an open neighbourhood of$\,\;X\cup K\;$
and $\mathfrak{V}$ is an open cover of $M$. As $K$ is closed in $M$, it is
paracompact in itself. As it is locally compact, $\;K=\bigcup_{\beta\in
B}K_\beta\,$, where $\,\{K_\beta:\beta\in B\}\,$ is a locally finite family
of compact sets in $M$ whose relative interiors cover $K$; the index set $B$
may be assumed not to contain $0$. By 5.1.19 of \cite{engelking}, there are
open sets $\;W_\beta\supseteq K_\beta\,$, for each $\;\beta\in B\,$, and
$\;W_0\supseteq X\,$, such that $\;\mathcal{T}:=\{W_\beta:\beta\in
B\cup\{0\}\}\;$ is locally finite in $M$. I may also suppose that
$\;W_\beta\subseteq V\;$ for each$\,\;\beta\in B_0:=B\cup\{0\}\,$. Write
$\,K_0\,$ for $X$. \\
$\,$

\emph{(\addtocounter{SEQnorth}{1}\alph{SEQnorth}\label{secb})}%
\hspace*{0.5cm}If $\;x\notin X\cup K\,$, let $\,U(x)\,$ be an open
neighbourhood of $x$ in $M$ which does not meet $\,X\cup K\,$ and meets only
finitely many $\,W_\beta\,$. If $\;x\in X\cup K\,$, let \ 
\begin{equation*}
B(x):=\{\beta\in B_0:x\in K_\beta\}\,\text{,}
\end{equation*}
which is finite, and set $\;U(x):=\left(\bigcap_{\beta\in
B(x)}W_\beta\right)\setminus\left(\bigcup_{\gamma\in B\setminus
B(x)}K_\gamma\right)\,$, which is also an open neighbourhood of $x$. In this
way $\,U(x)\,$ has the property for each $x$ that it meets only finitely many
$\,W_\beta\,$, and, if it meets $\,K_\beta\,$, then $\;x\in K_\beta\;$ and
$\;U(x)\subseteq W_\beta\,$. 

Let $\;\mathcal{S}:=\{U(x):x\in M\}\,$, an open cover of $M$. Let
$\mathfrak{W}$ be an open cover of $M$ which simultaneously refines
$\mathfrak{V}$ and $\mathcal{S}$. By hypothesis, there is a
$\mathfrak{W}$-extension system $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$ in $M$
such that $\;X\subseteq U_0(\Theta)\;$ and $\;U(\Theta)\subseteq W\,$. \\
$\,$

\emph{(\addtocounter{SEQnorth}{1}\alph{SEQnorth}\label{secc})}%
\hspace*{0.5cm}For each $\;\beta\in B\,$, there is, as $\,K_\beta\,$ is
compact, a finite subset $\,\Phi(\beta)\,$ of $J$ such that
$\;K_\beta\subseteq\bigcup_{j\in\Phi(\beta)}U_0^j\,$. By omitting indices for
which it is false, I may suppose that 
\begin{equation}\label{meet}
\;U_0^k\cap K_\beta\neq\emptyset\qquad\text{for each }\;k\in\Phi(\beta)\,.
\end{equation}
\ \ Let $\;\Phi:=\bigcup_{\beta\in B}\Phi(\beta)\,$, and set
$\;\Theta':=\Theta\cup\Phi\,$. \\
$\,$

\emph{(\addtocounter{SEQnorth}{1}\alph{SEQnorth}\label{secd})}%
\hspace*{0.5cm}Take any $\;\psi\in J\setminus\Theta\,$. Then
$\;\{\theta\in\Theta:U^\psi\cap U^\theta\neq\emptyset\}\,$ is finite by
\ref{extfam}\emph{(ii)}. On the other hand, if $\;\phi\in\Phi\,$,
$\;U^\phi\subseteq U(x)\,$ for some $\;x\in M\;$ (as $\,\{U^j\}\,$ refines
$\mathcal{S}$); but, by (\ref{meet}), $\;U^\phi\cap K\neq\emptyset\,$, and
therefore $\;U(x)\cap K\neq\emptyset\,$. From \emph{(b)}, $\;U(x)\subseteq
W_\beta\;$ for each $\;\beta\in B\;$ such that $\;U(x)\cap
K_\beta\neq\emptyset\,$. Hence, if $\;\phi\in\Phi$($\beta$)$\,$,
$\;U^\phi\subseteq U(x)\subseteq W_\beta\,$, and, if $\;U^\psi\cap
U^\phi\neq\emptyset\,$, necessarily $\;U^\psi\cap W_\beta\neq\emptyset\,$.
But, again because $\,\{U^j\}\,$ refines $\mathcal{S}$, there are only
finitely many such $\beta$ (see \emph{(b)}). Ergo, $\phi$ must be in one of
finitely many finite sets $\,\Phi(\beta)\,$, and $\,\{\phi\in\Phi:U^\phi\cap
U^\psi\neq\emptyset\}\,$ is finite.\\
$\,$

\emph{(e)}\hspace*{0.5cm}The pair $\,(\{(U^j,U_0^j):j\in J\},\Theta')\,$
clearly satisfies \ref{extfam}\emph{(i)}. I have just shown that, if
$\;\psi\in J\setminus\Theta'\;$ (indeed, if $\;\psi\in J\setminus\Theta\,$),
$\;U^\psi\cap U^j\neq\emptyset\;$ for only finitely many $\;j\in\Theta'\,$,
which is \ref{extfam}\emph{(ii)}. As for \ref{extfam}\emph{(iii)} and
\ref{strong}\emph{(iv}$'$\emph{)}, they are evidently still true. \emph{ }
\end{proof}

This result would remain true if \ref{extsys} were substituted for
\ref{strong} in the definition \ref{extensible} (just read
\ref{extsys}\emph{(iv)} for \ref{strong}\emph{(iv}$'$\emph{)}). Note that
$\mathcal{D}$ need only be applicable (cf. \ref{submanif}). However, even the
null set has not yet been proved to be simply $\mathcal{D}$-extensible.

One might on instinct expect a converse to the Proposition, stating that, if
$\,X\cup K\,$ is simply $\mathcal{D}$-extensible, $X$ is closed, and $K$ is
closed locally compact in $M$, then $X$ is simply $\mathcal{D}$-extensible.
This is not obvious, and probably false for the reasons noted after
\ref{extensible}.

\begin{lemma}
\ L\label{locquot}et $\mathcal{D}$ be an adequate category (see\emph{
\ref{adequate}}), and let $M$ be a $\mathcal{D}$-manifold, furnished with a
Finsler structure $p$ and induced metric $d$ and modelled on the normed space
$E$, with a $\mathcal{D}$-submanifold $N$ modelled on the closed linear
subspace $F$ of $E$ (see\emph{ \ref{subman}}). Metric balls are with
reference to the metric $d$. Suppose $\;x\in N\,$, and
$\;\omega:B(x;\frac{24}{25}\rho_{1/5}(x))\longrightarrow E\;$ is a
$\frac{1}{5}$-flat $\mathcal{D}$-submanifold chart for $N.\label{hh}$
$\omega$ and $p$ induce a norm $\,\left\|\,\right\|_{\omega,x}\,$ on $E$, as
at \emph{(\ref{normdef})} of\emph{ \ref{omisery}}.
Let$\,\;\pi:E\longrightarrow E/F\;$ be the quotient map, and let
$\,\left\|\,\right\|_{\omega,x}''\,$ denote the quotient norm on $\,E/F\,$.
Then, if $\;y\in B(x;\frac{1}{6}\rho_\epsilon(x))\,$, \ 
\begin{equation}\label{reldist}
\textstyle\frac{5}{6}\left\|\pi\circ\omega(y)\right\|_{\omega,x}''\leq
d(y,N)\leq\frac{6}{5}\left\|\pi\circ\omega(y)\right\|_{\omega,x}''\,\text{,}
\end{equation}
where, as usual, $\;d(y,N):=\inf\{d(y,z):z\in N\}\,$ (and similarly for other
subsets of $M$).
\end{lemma}

\begin{proof}
For notational convenience, write $\;\epsilon:=\frac{1}{5}\,$. Recall from
\ref{omisery} that $\,B_{\omega,x}'(\eta;\kappa)\,$ denotes a ball in $E$
with respect to $\,\left\|\xi\right\|_{\omega,x}\,$. If $\;y\in
B(x;\frac{1}{6}\rho_\epsilon(x))\,$, then
$\;d(y,N)<\frac{1}{6}\rho_\epsilon(x)\,$. Now,
$\;d(y,z)>\frac{1}{6}\rho_\epsilon(x)\;$ for any $\;z\notin
B(x;\frac{1}{3}\rho_\epsilon(x))\,$, and therefore $\,d(y,N)\,$ may be
calculated as the infimum of the distances $\,d(y,z)\,$ for points $\;z\in
N\cap B(x;\frac{1}{3}\rho_\epsilon(x))\,$. For any such $z$, \ref{distances}
implies that
$\;\left\|\omega(y)-\omega(z)\right\|_{\omega,x}\leq\textstyle\frac{6}{5}d(y,%
z)\,$. Taking infima over all choices of $z$, I find that
$\;\left\|\pi\circ\omega(y)\right\|_{\omega,x}''\leq\frac{6}{5}d(y,N)\,$, the
first inequality of (\ref{reldist}). 

By \ref{omisery}, $\;\omega(B(x;\frac{1}{6}\rho_\epsilon(x))\subseteq
B_{\omega,x}'(\omega(x);\frac{1}{5}\rho_\epsilon(x))\,$. As a consequence, 
\begin{equation*}
\begin{aligned}[t]
\left\|\pi\circ\omega(y)\right\|_{\omega,x}''&<\textstyle\frac{1}{5}\rho_%
\epsilon(x)\,.
\end{aligned}
\end{equation*}
\ If $\;\zeta\in F\;$ and, for some positive $\delta$, \ 
\begin{equation*}
\textstyle\left\|\omega(y)-\zeta\right\|_{\omega,x}<\left\|\pi\circ\omega(y)%
\right\|_{\omega,x}''+\delta<\frac{1}{5}\rho_\epsilon(x)\,\text{,}
\end{equation*}
then $\;\left\|\zeta\right\|_{\omega,x}<\frac{2}{5}\rho_\epsilon(x)\,$, and,
once again applying to \ref{omisery}, $\;\zeta=\omega(w)\;$ for some $\;w\in
N\cap B(x;\frac{12}{25}\rho_\epsilon(x))\,$. By \ref{distances}, 
\begin{equation*}
d(y,N)\leq
d(y,w)\leq\textstyle\frac{6}{5}\left\|\omega(y)-\zeta\,\right\|_{\omega,x}%
\leq\frac{6}{5}\left(\left\|\pi\circ\omega(y)\right\|_{\omega,x}''+\delta%
\right)\,.
\end{equation*}
Since $\delta$ is arbitrarily small,
$\;d(y,N)\leq\frac{6}{5}\left\|\pi\circ\omega(y)\right\|_{\omega,x}''\,$,
which is the second inequality of (\ref{reldist}).
\end{proof}

\begin{proposition}
\label{submanif}Let M be a separable metrizable $\mathcal{D}$-manifold
modelled on the normed space $E$, where $\mathcal{D}$ is an adequate
category. \ Let $\,\{N_\beta:\beta\in B\}\,$ be a discrete family of
$\mathcal{D}$-submanifolds of $M$, where, for each $\;\beta\in B\,$,
$\,N_\beta\,$ is modelled on the closed linear subspace $\,F_\beta\,$ of $E$
. Then $\;N:=\bigcup_{\beta\in B}N\,$ is a simply $\mathcal{D}$-extensible
subset of $M$.
\end{proposition}

\begin{proof}
Since $M$ is paracompact, it admits a Finsler structure $p$ which, in turn,
induces a metric $d$ (with respect to which metric balls are defined).
Suppose given an open cover $\mathfrak{V}$ of $M$ and an open neighbourhood
$V$ of $N$ in $M$. Let $\mathfrak{A}$ be the hereditary $\mathcal{D}$-atlas
for $M$ consisting of $\mathcal{D}$-charts $\;\phi:U\longrightarrow E\;$ such
that 
\begin{equation*}
\begin{aligned}[t]
&\text{\emph{(i)}}&\hspace{2mm}&\,U\cap N_\beta\neq\emptyset\;\text{ for at
most one index }\beta\text{,}\\
&\text{\emph{(ii)}}&\hspace{2mm}&\phi\text{ is a
}\mathcal{D}\text{-submanifold}\,\text{chart for }\,N_\beta\,\text{, for all
}\;\beta\in B\,\text{,}\\
&\text{\emph{(iii)}}&\hspace{2mm}&\text{there is a
}\;W\in\mathfrak{V}\;\text{ such that either }\;U\subseteq W\cap V\;\text{ or
}\;U\subseteq W\setminus N\,.
\end{aligned}
\end{equation*}
Since $\,\{N_\beta\}\,$ is a discrete family, these charts do form an atlas.
Indeed, if $\;x\notin N\,$, take any $\mathcal{D}$-chart about $x$, and
restrict it to an open neighbourhood $U$ of $x$ such that $\;U\cap
N=\emptyset\;$ and $\;U\subseteq W\;$ for some $\;W\in\mathfrak{V}\,$; if
$\;x\in N_\beta\,$, take any $\mathcal{D}$-submanifold chart for
$\,N_\beta\,$ about $x$ and restrict its domain to an open neighbourhood $U$
of $x$ such that $\;U\cap N_\gamma=\emptyset\;$ for any $\;\gamma\neq\beta\;$
and $\;U\subseteq W\cap V\;$ for some $\,W\in\mathfrak{V}\,$.

As in \ref{locquot}, take $\;\epsilon:=\frac{1}{5}\,$, and let
$\,\rho_\epsilon\,$ be the modulus of $\epsilon$-flatness with respect to $p$
and $\mathfrak{A}$ (see \ref{modflat}).\\
$\,$

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec})}\hspace*{0.5cm}The open cover
$\,\{B(x;\frac{1}{120}\rho_\epsilon(x)):x\in N\}\,$ of $N$ has a countable
subcover, $\,\{B(x_n;\frac{1}{120}\rho_\epsilon(x_n)):n\in\mathbb{N}\}\,$.
For each $\;n\in\mathbb{N}\,$, let \ 
\begin{equation}\label{plunder}
\left\{
\begin{aligned}
U_0^n&:=B(x_n;\textstyle\frac{1}{120}\rho_\epsilon(x_n))\cap\{y\in
M:d(y,N)<\textstyle\frac{1}{240}n^{-1}\rho_\epsilon(x_n)\}\,\text{,}\\
\displaystyle\textstyle
U^n&:=B(x_n;\textstyle\frac{1}{60}\rho_\epsilon(x_n))\cap\{y\in
M:d(y,N)<\textstyle\frac{1}{120}n^{-1}\rho_\epsilon(x_n)\}\,.
\end{aligned}
\right.
\end{equation}
Since $\;B(x_n;\frac{1}{120}\rho_\epsilon(x_n))\cap N=U_0^n\cap N\,$,
$\,\{U_0^n\}\,$ remains an open cover of $N$. By the construction of
$\mathfrak{A}$ and $\,\rho_\epsilon\,$, either
$\;B(x_n;\frac{1}{2}\rho_\epsilon(x_n))\subseteq M\setminus N\;$ (which is
false) or $\,B(x_n;\frac{1}{2}\rho_\epsilon(x_n))\subseteq V\;$ for each $n$,
so that $\;\bigcup_{n\in\mathbb{N}}U^n\subseteq V\,$, as required by
\ref{extensible}.

$N$ being closed, $\;d(x,N)>0\;$ for each $\;x\in M\setminus N\,$. Cover
$\,M\setminus N\,$ by the balls
$\,B(x;\frac{1}{120}\min(\rho_\epsilon(x),d(x,N)))\,$, where $x$ varies over
$\,M\setminus N\,$. There is a countable subcover, whose index set $\Psi$ may
be chosen to be disjoint from $\mathbb{N}$,
$\,\{B(x_\psi;\frac{1}{120}\min(\rho_\epsilon(x_\psi),d(x_\psi,N))):\psi\in%
\Psi\}\,$; define $\;\Theta:=\mathbb{N}\,$, $\;J:=\Theta\cup\Psi\,$, so that
$\;x_j\in N\;$ when and only when $\;j\in\Theta\,$, and set, for each
$\;\psi\in\Psi\,$,
\begin{equation}\label{cover}
\left\{
\begin{aligned}
\textstyle
U_0^\psi&:=B(x_\psi;\textstyle\frac{1}{120}\min(\rho_\epsilon(x_\psi),d(x_%
\psi,N)))\,,\\
\textstyle
U^\psi&:=B(x_\psi;\textstyle\frac{1}{60}\min(\rho_\epsilon(x_\psi),d(x_\psi,%
N)))\,.
\end{aligned}
\right.
\end{equation}
The condition \ref{extfam}\emph{(i)} is fulfilled, and, for any $\;j\in J\,$,
$\;U^j\subseteq B(x_j;\frac{1}{60}\rho_\epsilon(x_j))\,$.\\
\ 

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec})}\hspace*{0.5cm}Given
$\;\psi\in\Psi\,$, (\ref{cover}) shows that, for any $\;z\in U^\psi\,$, 
\begin{equation*}
d(z,N)\geq
d(x_\psi,N)-\textstyle\textstyle\frac{1}{60}d(x_\psi,N)=\frac{59}{60}d(x_\psi,%
N)\,.
\end{equation*}
On the other hand, if $\;z\in U^n\,$, (\ref{plunder}) implies that
$\;d(z,N)<\frac{1}{120}n^{-1}\,$, since $\;\rho_\epsilon(x_n)\leq 1\;$ by
definition \ref{modflat}. Hence, $\;U^\psi\cap U^n\neq\emptyset\;$ only if
$\;\frac{1}{120}n^{-1}>\frac{59}{60}d(y_\psi,N)\,$, and so for only finitely
many $n$. This proves \ref{extfam}\emph{(ii)}. \\
\ 

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec})}\hspace*{0.5cm}Suppose
that$\,\;\psi\in\Psi\,$. There are two possibilities. The first is that, for
any $\;r,s\in J\;$ such that $\;U^\psi\cap U^r\neq\emptyset\neq U^r\cap
U^s\,$, necessarily $\;s\in\Psi\,$. This means, in particular, that the sets
appearing in \ref{strong}\emph{(iv}$'$\emph{)(b)} are all metric balls of the
form (\ref{cover}). By \ref{hamilton}, $\;U^s\subseteq
B(x_\psi;\frac{1}{6}\rho_\epsilon(x_\psi))\,$. By the definition
\ref{modflat}, there is some $\epsilon$-flat $\mathcal{D}$-chart with domain
$\,B(x_\psi;\frac{2}{3}\rho_\epsilon(x_\psi))\,$, and I choose such a chart
as $\,\omega_\psi\,$, taking
$\;O_\psi:=B(x_\psi;\frac{2}{3}\rho_\epsilon(x_\psi))\,$. Then \
\ref{strong}\emph{(iv}$'$\emph{)(a) }is satisfied automatically. \ \\
\ 

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec}\label{estball})}%
\hspace*{0.5cm}Suppose in the case \emph{(c) }that $\;k,l\in J\;$ and
$\;U^\psi\cap U^k\neq\emptyset\neq U^k\cap U^l\,$. From \ref{meetingball}, \
$\;\rho_\epsilon(x_l)<\left(\frac{61}{59}\right)^2\rho_\epsilon(x_\psi)\,$.
Take in \ref{ball} $\;y:=x_l\,,$ $x:=x_\psi\,$, and suppose 
\begin{equation*}
\textstyle R:=\frac{2}{3}\rho_\epsilon(x_\psi)\,,\quad
r:=\frac{1}{6}\rho_\epsilon(x_\psi)\,,\quad
0<\delta\leq\frac{1}{120}\rho_\epsilon(x_l)\,.
\end{equation*}
The hypothesis of \ref{ball} is satisfied, because \ 
\begin{equation*}
\textstyle
2\left(\frac{6}{5}\right)^3\left(\frac{1}{6}+\left(\frac{61}{59}\right)^2%
\frac{1}{120}\right)\rho_\epsilon(x_\psi)\leq\frac{2}{3}\rho_\epsilon(x_\psi)%
\,.
\end{equation*}
So \ 
\begin{equation*}
\begin{aligned}[t]
&\textstyle
A(\frac{6}{5},\omega_\psi(x_l))\text{co}(\omega_\psi(B(x_l;\delta)))\subseteq%
\omega_\psi\left(B\left(x_l;\left(\frac{6}{5}\right)^3\delta\right)\right)%
\subseteq\omega_\psi(B(x_l;2\delta))\,.
\end{aligned}
\end{equation*}
If, in particular,
$\;\delta:=\frac{1}{120}\min(\rho_\epsilon(x_l),d(x_l,N))\,$, then \ 
\begin{equation*}
\textstyle
A(\frac{6}{5},\omega_\psi(x_l))\text{co}(\omega_\psi(U_0^l))\subseteq\omega_%
\psi(U^l)\,.
\end{equation*}

\ 

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec}\label{clam})}\hspace*{0.5cm}The
second possibility is that there are \emph{some }$\;r\in J\;$ and
$\;\theta\in\Theta\;$ such that $\;U^\psi\cap U^r\neq\emptyset\neq U^r\cap
U^\theta\,$. (Here, $\,U^\theta\,$ is \emph{not} defined as a metric ball;
see (\ref{plunder}). Indeed, (\ref{reldist}) shows it usually is not one). In
this case, I choose such a $\theta$ and $r$. Certainly $\;x_\theta\in
N_\beta\;$ for some $\;\beta\in B\,$. There is an $\epsilon$-flat
$\mathcal{D}$-submanifold chart for $\,N_\beta\,$ with domain
$\,B(x_\theta;\frac{24}{25}\rho_\epsilon(x_\theta))\,$; I take one such chart
to be $\,\omega_\psi\,$, letting $\,O_\psi\,$ be
$\,B(x_\theta;\frac{24}{25}\rho_\epsilon(x_\theta))\,$, which again satisfies
\ \ref{strong}\emph{(iv}$'$\emph{)(a)}. 

If $\;y\in B(x_\theta;\frac{1}{6}\rho_\epsilon(x_\theta))\,$, then
$\;d(y,N_\gamma)>\frac{5}{6}\rho_\epsilon(x_\theta)\;$ for any
$\;\gamma\neq\theta\,$, since, by the definition of $\,\rho_\epsilon\,$,
$\;B(x_\theta;d(y,x_\theta)+\frac{5}{6}\rho_\epsilon(x_\theta))\cap
N_\gamma=\emptyset\,$. Therefore, $\;d(y,N)=d(y,N_\beta)\,$.\\
$\,$

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec}\label{insert})}%
\hspace*{0.5cm}Suppose $\;k,l\in J\;$ and $\;U^\psi\cap U^k\neq\emptyset\neq
U^k\cap U^l\,$. By \ref{hamilton} (this time with a full chain of five sets)
$\;U^l\subseteq B(x_\theta;\frac{1}{6}\rho_\epsilon(x_\theta))\,$. If
$\;l\in\Psi\,$, note that 
\begin{equation}\label{weakestimate}
\textstyle
2\left(\frac{6}{5}\right)^3\left(\frac{1}{6}+\left(\frac{61}{59}\right)^4%
\frac{1}{120}\right)\rho_\epsilon(x_\psi)\leq\frac{24}{25}\rho_\epsilon(x_%
\psi)\,\text{,}
\end{equation}
\ so that \ref{ball} applies again (compare \emph{(d)}$)\,$and 
\begin{equation*}\label{hhh}
\textstyle
A(\frac{6}{5},\omega_\psi(x_l))(\text{co}(\omega_\psi(U_0^l))\subseteq\omega_%
\psi(U^l)\,.
\end{equation*}

\emph{(\addtocounter{SEQsec}{1}\alph{SEQsec})}\hspace*{0.5cm}If
$\;l\in\mathbb{N}=\Theta\,$, take
$\;\delta:=\frac{1}{120}\rho_\epsilon(x_l)\;$ \ in \ref{ball}, and one
deduces that \ 
\begin{equation}\label{shrinkball}
\left\{
\begin{aligned}
\textstyle
A(\frac{6}{5},\omega_\psi(x_l))\,\text{co}(\omega_\psi(U_0^l))&\subseteq
A(\textstyle\frac{6}{5},\omega_\psi(x_l))\,\text{co}(\omega_\psi(B(x_l;%
\textstyle\frac{1}{120}\rho_\epsilon(x_l)))\\
\textstyle&\subseteq\omega_\psi(B(x_l;\textstyle\frac{1}{60}\rho_%
\epsilon(x_l))\,.
\end{aligned}
\right.
\end{equation}
If $\;y\in U_0^l\,$, (\ref{reldist}) holds with $\,x_\theta\,$ for ``$x$'',
$\,F_\beta\,$ for ``$F$'', $\,\omega_\psi\,$ for ``$\omega$'', $\,N_\beta\,$
for ``$N$'', and the projection $\;\pi_\beta:E\longrightarrow E/F_\beta\;$
instead of ``$\pi$''. The hypotheses of \ref{locquot} are satisfied. From the
first inequality of (\ref{reldist}) and the definition (\ref{plunder}), in
which, from \emph{(e)}, one may read $\,d(y,N_\beta)\,$ for ``$d(y,N)$'', \ 
\begin{equation*}
\textstyle\left\|\pi_\beta\circ\omega_\psi(y)\right\|_{\omega_\psi,x_%
\theta}''<\frac{1}{200}l^{-1}\rho_\epsilon(x_l)\,.
\end{equation*}
As$\,\,\pi_\beta\,$ is linear and $\;\pi_\beta\circ\omega_\psi(x_l)=0\,$,
this implies that 
\begin{equation}\label{shrinkdist}
\textstyle\left\{
\begin{aligned}
\textstyle\pi_\beta(A(\frac{6}{5},\omega_\psi(x_l))\,\text{co}(\omega_%
\psi(U_0^l)))&=A(\textstyle\frac{6}{5},0)\,\text{co}(\pi_\beta(\omega_%
\psi(U_0^l)))\\
=\textstyle\frac{6}{5}\text{co}(\pi_\beta(\omega_\psi(U_0^l)))&\subseteq
B_{\omega_\psi,x_\theta}''(0;\textstyle\frac{3}{500}l^{-1}\rho_\epsilon(x_l))%
\,\text{,}
\end{aligned}
\right.
\end{equation}
the$\,\,\left\|\,\right\|_{\omega_\psi,x_\theta}''$-ball about the origin in
$E$ of radius $\,\frac{3}{500}l^{-1}\rho_\epsilon(x_l)\,$. 

Because of the second inequality of (\ref{reldist}), if $\;y\in
B(x_\theta;\frac{1}{6}\rho_\epsilon(x_\theta))\;$ and
$\;\left\|\pi_\beta\circ\omega_\psi(y)\right\|_{\omega_\psi,x_\theta}''<%
\frac{3}{500}l^{-1}\rho_\epsilon(x_l)\,$, then 
\begin{equation*}
\textstyle
d(y,N_\beta)<\frac{9}{1250}l^{-1}\rho_\epsilon(x_l)<\frac{1}{120}l^{-1}\rho_%
\epsilon(x_l)\,.
\end{equation*}
\ Hence, by (\ref{shrinkdist}), (\ref{shrinkball}) and (\ref{plunder}), and
because $\,\omega_\psi\,$ is one-one on
$\,B(x_\theta;\frac{1}{6}\rho_\epsilon(x_\theta))\,$,
\begin{equation*}
\begin{aligned}[t]
\textstyle A(\frac{6}{5},\omega_\psi&(x_l))\,\text{co}(\omega_\psi(U_0^l))\\
&\subseteq\omega_\psi(B(x_l;\textstyle\frac{1}{60}\rho_\epsilon(x_l))\cap\{y%
\in N:d(y,N)<\textstyle\frac{1}{120}l^{-1}\rho_\epsilon(x_l)\})\\
&=\omega_\psi(U^l)\,.
\end{aligned}
\end{equation*}
Since $\;$co$(\omega_\psi(U_0^l))\,$ is open, \ref{rk} shows that
$\;$co$(\omega_\psi(U_0^l))\prec\omega_\psi(U^l)\,$. I have now established
\ref{strong}\emph{(iv}$'$\emph{)(b)} in all cases. \label{move}Finally,
\ref{extfam}\emph{(iii)} follows from (\ref{plunder}) and (\ref{cover}). \
\label{h}
\end{proof}

\begin{remark}
\ The above method of proof may be extended, with some additional
complications, to show that certain other sets in $M$ are also simply
$\mathcal{D}$-extensible: for instance, $\mathcal{D}$-submanifolds with
corners and locally finite arrays of $\mathcal{D}$-submanifolds in ``general
position'', as suitably defined. The above case is probably the most
interesting. It should be noted that the submanifolds need not be
split-embedded. Recall, however, that the extension problem concerns
functions already defined on a neighbourhood of the supposed core.

The numerical values used in the proof are of course without significance.

The construction fulfils the condition that $\;\Theta=\{j\in J:U^\theta\cap
N\neq\emptyset\}\,$, which is a natural property to ask of a
$\mathfrak{V}$-extension system; it was not, however, required in the
extension argument \ref{prop}. If it were imposed, \ref{extensible} could be
simplified --- see \ref{definition}.
\end{remark}

\begin{theorem}
\ L\label{theorem}et $\mathcal{C}$ be an admissible category (see\emph{
\ref{admiss}}) of mappings between topological vector spaces, and let
$\mathcal{C}'$ be the subcategory of $\mathcal{C}$ whose objects are normed
spaces and whose morphisms are \emph{C}$^1$\emph{ }in the sense of
Fr\a'echet. Let $M$ be a separable metrizable
$\mathcal{L}\mathcal{C}'$-manifold. Suppose that $X$ is a subset of $M$ that
may be expressed as the union of a discrete family of
$\mathcal{L}\mathcal{C}'$-submanifolds and of a closed locally compact
subset. Then, to any open neighbourhood $U$ of $X$ in $M$, there exists a
neighbourhood $V$ of $X$, with $\;V\subseteq U\,$, such that

\emph{(i)}\hspace*{0.5cm}~~for any locally trivial
$\mathcal{L}\mathcal{C}$-fibration $\;\pi:L\longrightarrow M\;$ of
$\mathcal{L}\mathcal{C}$-manifolds, with fibre $N$ that is
$\mathcal{L}\mathcal{C}$-diffeomorphic to an open convex set $Y$ in the
topological vector space $F$, and to any $\mathcal{L}\mathcal{C}$-section
$\sigma$ of $\pi$ over $U$, there corresponds an
$\mathcal{L}\mathcal{C}$-section $\tau$ of $\pi$ over $M$ such that
$\;\sigma\mid V=\tau\mid V\,$; call $\tau$ the ``extension'' of $\sigma$;

\emph{(ii)}\hspace*{0.5cm}~if $\pi$ is an $\mathcal{L}\mathcal{C}$-vector
fibration (see\emph{ \ref{product}}), and a linear space $\mathcal{F}$ of
$\mathcal{L}\mathcal{C}$-sections of $\pi$ over $U$ is given (with respect to
pointwise operations), then the extensions of these sections may be so
constructed as to define a linear operator from $\mathcal{F}$ into the space
of $\mathcal{L}\mathcal{C}$-sections of $\pi$ over $M$;

\emph{(iii)}\hspace*{0.5cm}for any $\mathcal{L}\mathcal{C}$-mapping
$\;g:U\longrightarrow F\;$ into a topological vector space $F$, there is an
$\mathcal{L}\mathcal{C}$-mapping $\;f:M\longrightarrow F\;$ (the
``extension'' of $g$) such that $\;f\mid V=g\mid V\;$ and
$\;f(M)\subseteq\,$\emph{co}$(g(U))\,$; furthermore, if a linear space
$\mathcal{F}$ of $\mathcal{L}\mathcal{C}$-mappings $\;U\longrightarrow F\;$
(with respect to pointwise operations) is given, their extensions may be so
constructed as to define a linear operator from $\mathcal{F}$ into the space
of \ $\mathcal{L}\mathcal{C}$-mappings $\;M\longrightarrow F\,$. 
\end{theorem}

As remarked at \ref{remark}, the topology on $F$ is not really important in
\emph{(iii)}.

\begin{proof}
$X$ is simply $\mathcal{L}\mathcal{C}'$-extensible by \ref{submanif} and
\ref{loccpctunion}; it follows that it is simply
$\mathcal{L}\mathcal{C}$-extensible, and the result follows by
\ref{coreexten} and the remark following it.
\end{proof}

\begin{corollary}
\ Any \label{end}paracompact \emph{C}$^\infty$ manifold $M$ modelled on the
Banach space $C$ or on the Banach space $\ell^1$ admits a nonconstant
real-valued \emph{C}$^\infty$ function.
\end{corollary}

\begin{proof}
It suffices to assume $M$ is connected, and therefore separable and
metrizable. By \cite{atkin2}1.3, $M$ is an $\mathcal{L}$C$^\infty$-manifold.
For distinct points $\;a,b\in M\,$, let $g$ take the values $1$ on a
neighbourhood of $a$ and $0$ on a neighbourhood of $b$. By \ref{theorem},
there is an $\mathcal{L}$C$^\infty$ function
$\;f:M\longrightarrow\mathbb{R}\;$ which agrees with $g$ on some (smaller)
neighbourhoods of $a$ and $b$.
\end{proof}

$\,$

\section{Other cases.}

\S\S 4 and 5 rely on continuous Fr\a'echet differentiation in normed spaces
to satisfy the approximate convexity requirements of \ref{strong} for a
$\mathcal{D}$-manifold. If the model is not normed and the differential
calculus employed does not have suitable properties with respect to convex
neighbourhoods of the origin (properties that one has no right to expect),
arguments such as those must fail. However, \ref{prop} may still sometimes be
used, in a slightly modified version. Instead of introducing an ``approximate
convex structure'' on the manifold by means of a Finsler structure, one may
have something given \emph{a priori}. The simplest example is an open set in
a locally convex space, but a little more generality is possible. 

\begin{theorem}
\label{badjob} Let $X$ be a subset of a locally convex Hausdorff Lindel\"of
space $E$, and let $Y$ be a convex subset of a vector space $F$; let
$\mathcal{F}$ be a $Y$-admissible class of functions $\;X\longrightarrow Y\;$
\ (see \cite{atkin2}\emph{1.7}). If $K$ is a subset of $X$ that is closed
locally compact in $E$ and $U$ is an open neighbourhood of $K$ in $E$, there
is an open neighbourhood $\;V\subseteq U\;$ of $K$ in $E$ such that, for any
$\mathcal{L}\mathcal{F}$-function $\;g:U\cap X\longrightarrow Y\,$, there is
an $\mathcal{L}\mathcal{F}$-function $\;f:X\longrightarrow Y\;$ for which
$\;f\mid V\cap X=g\mid V\cap X\,$. ($f$ may be called the ``extension'' of
$g$). Moreover, if a whole convex set $\mathcal{E}$ of
$\mathcal{L}\mathcal{F}$-functions $\;U\cap X\longrightarrow Y\;$ (with
respect to pointwise operations) is given, their extensions may be so
constructed as to define an affine-linear map from $\mathcal{E}$ into the
space of $\mathcal{L}\mathcal{F}$-functions $\;X\longrightarrow Y\,$.
\end{theorem}

\begin{proof}
The details of this proof will be omitted, for it contains no new ideas. As
in \ref{locquot}, one may construct a $\mathfrak{V}$-extension family for $X$
of the form $\,(\{(U^j,U_0^j):j\in J\},\Theta)\,$, where 
\begin{equation*}
\textstyle U^j:=X\cap(x_j+B_j)\,\text{,}\quad
U_0^j:=X\cap(x_j+\frac{1}{2}B_j)\,\text{,}\quad\mathfrak{V}:=\{U\cap
X,X\setminus K\}\,\text{,}
\end{equation*}
$\,B_j\,$ an open convex neighbourhood of $0$ in $E$, $\;\Theta:=\{j\in
J:U^j\cap K\neq\emptyset\}\,$; one takes 
\begin{equation*}
V:=\bigcup_{\theta\in\Theta}U_0^\theta\subseteq\bigcup_{\theta\in\Theta}U^%
\theta\subseteq U\,.
\end{equation*}
The argument of \ref{prop} may now be followed, with many simplifications,
using \cite{atkin2}5.1 exactly as stated (with the set $X$); instead of
charts $\,\omega_j\,$, one has convex subsets of $E$. \ \ \ 

\end{proof}



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\end{document}
