Missing hypothesis in the equivalence between functionally structured spaces and smooth $n$-manifolds

Asked Jul 05 '21 at 22:34

Active Feb 07 '22 at 11:31

Viewed 69 times

In Problem 1 [Bredon, Topology and Geometry, Chapter II] it reads that a functional structure $F$ on a second countable Hausdorff space defines an $n$-manifold if, and only if, there is a cover with open sets $U$ and functions $f_1,\dots,f_n\in F(U)$ such that $$ g\in F(U)\iff \textrm{there exists $h$ smooth with }g = h\circ(f_1,\dots,f_n). $$ To me the case where $F(U)$ is the set of constant functions is a counterexample for the if part.

Am I missing anything? If not, which additional hypothesis should we add?

A functionally structured space is a topological space $(X,{\cal T})$ along with an assignment $U\mapsto F(U)$ for $U\in\cal T$, where $F(U)$ is subalgebra of $C(X,\mathbb R)$ that satisfies

$F(U)$ includes all constant functions;
$V\subseteq U$ and $f\in F(U)\implies f|_V\in F(V)$, and
$f\colon U\to\mathbb R$, $f|_{U_i}\in F(U_i)$ for all $i\in I$ and $U=\bigcup_{i\in I} U_i \implies f\in F(U)$.

edited Feb 07 '22 at 11:31

Martin Sleziak

53,687

asked Jul 05 '21 at 22:34

Leandro Caniglia

1,880

1

What is a functional structure? – Federico Fallucca Jul 05 '21 at 22:35
@FedericoFallucca I've added the definition. – Leandro Caniglia Jul 05 '21 at 22:41
A functional structure is not a space, as defined, so how can it be a manifold? – Thomas Andrews Jul 05 '21 at 22:49
@ThomasAndrews My bad. I've rewritten the problem. – Leandro Caniglia Jul 05 '21 at 22:53
1

It does look like there is something missing. Perhaps that $h$ must be unique? That would mean that the each $f_i$ would have to have all of $\mathbb R$ as the range. (Or perhaps just a dense subset of $\mathbb R?$) – Thomas Andrews Jul 05 '21 at 22:57
2

Strictly speaking, the constant functions don't satisfy item 3 in the definition of functionally structured space. If $U$ is the union of two disjoint open subsets and $f$ is constant on each of those subsets but with different values, then 3 is violated. But locally constant functions work, so you're right that more hypotheses are needed. – Andreas Blass Jul 05 '21 at 23:05
Please, use descriptive titles. "Missing hypothesis?" tells nothing about the subject of the question. – jjagmath Jul 06 '21 at 00:23
@jjagmath Thanks for the indication. – Leandro Caniglia Jul 06 '21 at 00:51

Missing hypothesis in the equivalence between functionally structured spaces and smooth $n$-manifolds

0 Answers0

Linked