I am trying to prove the following theorem:
Theorem. Every compact manifold is homeomorphic to a subset of some Euclidean space.
The manifolds I'm considering are the most general (without any smooth, PL or any structures). Here's a formal definition:
$n$–Manifolds. An $n$-manifold $\mathcal M$ is a second-countable, Hausdorff topological space such that every point $p\in\mathcal M$ has a neighborhood $U\subseteq\mathcal M$ that is homeomorphic to $\mathbb R^n$.
Here's my (incomplete) attempt at a proof:\
Proof. Suppose $\mathcal M$ is a compact $n$-manifold. By compactness, $\mathcal M$ has a finite open cover, say, $U_1,\dots,U_k$. By definition, we are guaranteed that each of these open subsets is homeomorphic to $\mathbb R^n$. That is, we can choose homeomorphisms $\phi_i:U_i\to\mathbb R^n$ for each $i$. We recall that since $\mathcal M$ is compact, it is also paracompact and hence admits a partition of unity subordinate to this open cover, say, $(\psi_i)_i$. Then I'm lost…
My idea was to use the functions $\phi_i$'s and $\psi_i$'s to construct a homeomorphism $\Phi:\mathcal M\to\mathbb R^j$ for some $j$ using the Pasting Lemma (also known as Gluing Lemma). But I don't know how to use that lemma to construct $\Phi$ and show that is it a homeomorphism. I have already spent over two days in trying to figure this out. Any help would be appreciated. TIA.
Best Answer
Take open finite open cover of $X$ by sets homeomorphic to $\mathbb{R}^n$, say $U_1, ..., U_k$. For each $i$, let $f_i:U_i\to \mathbb{R}^n$ be a homeomorphism. By considering the one-point compactification $\mathbb{R}^n\cup\{\infty\}\cong S^n$ of $\mathbb{R}^n$, you can extend $f_i$ to a continuous map $f_i:X\to \mathbb{R}^n\cup\{\infty\}$ where $f_i$ sends $X\setminus U_i$ to $\infty$. Verify this.
The map $f(x) = (f_1(x), ..., f_k(x))$ is then an embedding, since it's a closed injective continuous map. As mentioned above, the one-point compactification of $\mathbb{R}^n$ naturally embedds into $\mathbb{R}^{n+1}$, so this gives an embedding of $X$ into $\mathbb{R}^N$ for some $N\in\mathbb{N}$.