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I'm reading through Introduction to Smooth Manifolds by Lee and I've been stuck on part (b) of the following exercise for a while now:

Proposition 1.38.

Let $M$ be a topological $n$-manifold with boundary.

(a) Int $M$ is an open subset of $M$ and a topological $n$-manifold without boundary.

(b) $\partial M$ is a closed subset of $M$ and a topological $(n-1)$-manifold without boundary.

(c) $M$ is a topological manifold if and only if $\partial M = \emptyset$.

(d) If $n = 0$, then $\partial M = \emptyset$ and $M$ is a $0$-manifold.

Exercise 1.39. Prove the preceding proposition. For this proof, you may use the theorem on topological invariance of the boundary when necessary. Which parts require it?

(Here $\partial M$ denotes the manifold boundary of $M$, rather than the topological boundary.)

The theorem on topological invariance of the boundary states that $\partial M = (Int M)^c$. Showing that $\partial M$ is closed is easy enough if we use the theorem freely, but I'm interested in the question of whether the theorem is required. Denoting the theorem on topological invaraince by $T$, can we say that the proof of part (b) requires $T$ if $(b) \implies T$? If so, then I think we can say that the proof of part (b) does not require $T$, because the condition that $\partial M$ is closed (and an $(n-1)$-manifold) seems a lot weaker than the result $\partial M = (Int M)^c$.

However, I'm struggling with actually proving (b) without using $T$. I've tried all the standard ways to show a set is closed - showing the complement is open (by showing $(\partial M)^c$ is a union of open sets, finite intersection of open sets, or showing every point has a neighbourhood contained in the complement), showing $\partial M$ is an intersection of closed sets (which is of course equivalent to showing $(\partial M)^c$ is a union of open sets) or a finite union of closed sets, and showing $\partial M$ contains its topological boundary. The main problem I'm running into is that in order for a point $p$ to be in $\partial M$, we only need $\varphi(p) \in \partial \mathbb{H}^n$ for some coordinate chart $\varphi$, which is to say, $\partial M = \bigcup_{\varphi} \varphi^{-1}(\partial \mathbb{H}^n)$ where the union ranges over all coordinate charts $\varphi$. If I can reduce this to a finite union then I'm good (since each $\varphi$ is a homeomorphism and therefore a closed map), and I've been hoping that paracompactness would play a role in this somehow.

So I guess my questions are:

(1) Is it possible to prove this result without the theorem on topological invariance of the boundary?

(2) If so... how?

(3) If not, can I formally prove that the result requires the theorem on topological invariance of the boundary?

isolar.ii
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    When you work with smooth (or $C^1$) manifolds, the inverse function theorem does the heavy lifting, but for topological manifolds you need T to know that a homeomorphism must send boundary to boundary. – Ted Shifrin Dec 23 '24 at 00:34
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    Presumably you have in mind some (unstated) proof of the theorem on topological invariance of the boundary. That being so, in any later proof where you are tempted to apply that theorem, here's what you do instead: don't apply the theorem, instead just insert the proof of the theorem. Voila, you have rewritten your proof so that it does not apply the theorem on topological invariance of the boundary. – Lee Mosher Dec 23 '24 at 02:46
  • @LeeMosher Haha, that would be cheating. Also, the textbook says that the proof requires more machinery than we have available at this point, and defers the proof to a later chapter. So unless I want to develop the theory of De Rham cohomology to prove that the boundary of a manifold is closed, I'm stuck looking for ways to do it without topological invariance of the boundary. – isolar.ii Dec 23 '24 at 23:36
  • The first non trivial instance of this sort of result is in dimension 1. How can you prove that $[0,1[$ is not homeomorphic to $]0,1[$ , or that any homeomorphism of $[0,1[$ sends $0$ to $0$. – Thomas Dec 24 '24 at 10:56
  • I haven't read Lee's Smooth Manifolds but Topological Manifolds. 1.38(b) is Problem 3-1 in Lee's ITM. The $(n-1)$-manifold part can be proved without the invariance of the boundary, but the closed subset part needs it. See https://math.stackexchange.com/questions/554156/the-boundary-of-an-n-manifold-is-an-n-1-manifold – Duong Ngo Dec 25 '24 at 04:43

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