# Functionally structured spaces and manifolds

The question requires some definitions that I have listed below for your convenience. They can be found in Chapter VI, pages 297-298 of Bredon's Introduction to Compact Transformation Groups.

On a topological space $X$ a functional structure $F_{X}$ is an assingment defined on the collection of open sets $U\subset X$ taking $U\mapsto F_{X}(U)$ such that:

1. $F_{X}(U)$ is a subalgebra of the algebra of continuous real valued functions on $U$ and contains all constant functions.

2. If $V$ is open, $V\subset U$ and $f\in F_{X}(U)$ then $f|_{V}\in F_{X}(V)$.

3. If $\{U_{i}\}$ is a collection of open sets and $f|_{U_{i}}\in F_{X}(U_{i})$ for all $i$ then $f\in F_{X}(\bigcup_{i}U_{i})$.

The pair $(X,F_{X})$ is called a functionally structured space. E.g. $(\mathbb{R}^{n},C^{\infty})$, where $\mathbb{R^{n}}$ has the usual topology and $C^{\infty}(U)=\text{all$C^{\infty}$functions on$U$.}$

If $(X,F_{X})$ is a functionally structured space and $U\subset X$ is open then for $V\subset U$ open we define $F_{U}(V)=F_{X}(V)$ so that $(U,F_{U})$ is a functionally structured space.

A morphism of functionally structured spaces $(X, F_{X})$ and $(Y,F_{Y})$ is a continuous map $\varphi:X\rightarrow Y$ such that for any open set $V\subset Y$ and $f\in F_{Y}(V)$ we have $f\circ \varphi\in F_{X}(\varphi^{-1}(V))$. A morphism $\varphi$ is an isomorphism if $\varphi^{-1}$ exists as a morphism. We then say that $(X,F_{X})\simeq (Y,F_{Y})$.

An $n$-dimensional differentiable manifold is a second countable functionally structured Hausdorff space $(M,F)$ with the property that each point in $M$ has an open neighborhood $U$ such that $(U,F_{U})\simeq(V,C^{\infty}_{V})$ for some open set $V\subset \mathbb{R}^{n}$.

Question: Let $(X,F_{X})$ be a functionally structured space with the following property. Each point in $X$ has a neighborhood $U$ such that there are functions $f_{1},\ldots ,f_{n}\in F_{X}(U)$ verifying: A real valued function $g$ on $U$ is in $F_{X}(U)$ iff there exits a smooth real valued function $h$ of $n$ real variables such that $g(p)=h(f_{1}(p),\ldots,f_{n}(p))$ for all $p\in U$.

Does it follow that $\mathbf{(X,F_{X})}$ is an $\mathbf{n}$-dimensional differentiable manifold? (In the sense of the definition given above).

Edit: Sheaves are new to me, but after reading the comments I wanted to point out this and this MSE posts, and the wikipedia page and subwiki page where sheaves are used to define differentiable manifolds.

Edit: Sheaves my be used to define differentiable manifolds: Roughtly speaking, a differentiable manifold is a second countable Hausdorff topological space equipped with a subsheaf of the sheaf of continuous real valued function that is locally isomorphic to the sheaf of smooth real valued functions on some $\mathbb{R}^{n}$.

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What you call a "functional structure" is more commonly known as a "subsheaf of the sheaf of continuous real-valued functions". Your question is local and can be phrased without recourse to such complications, though. I think you should probably add some other hypotheses: for example, if $X$ is a one-point space then it automatically satisfies your condition. Perhaps you should ask that the smooth function $g$ be unique. –  Zhen Lin Aug 8 '13 at 18:47
I guess you mean that smooth function $h$ be unique, as $g$ is not assumed smooth. –  John Aug 8 '13 at 18:52
@ZhenLin It looks slightly more restrictive than a general sheaf in that we ask that all constant functions are their. –  Baby Dragon Aug 8 '13 at 19:33
Well, to be more precise, it has to be a subsheaf considered as a sheaf of $\mathbb{R}$-algebras, and as you say it has to be non-trivial if the space is non-empty. –  Zhen Lin Aug 8 '13 at 20:09

Consider $X=\mathbb{R}$ with the usual topology and define a functional structure $F_X$ on $X$ as follows: For any open interval $I\subset\mathbb{R}$ let $F_{X}(I)$ consist only of constant functions. Now, an arbitrary open set $U\subset\mathbb{R}$ can be uniquely written (i.e. up to a permutation of the intervals) as the disjoint union of countably many open intervals. Therefore, we can define $F_{X}(U)$ to be the set of all real-valued functions which are constant on open intervals building up $U$.
Note that $(X,F_{X})$ as defined satisfies the hypothesis of the question but it is not a smooth manifold.
Additional hypotheses are then needed for an affirmative answer. E.g. requiring that the function $(f_{1},\ldots, f_{n})$ is locally invertible.