Problem 1.3 (Topological manifolds are \(\sigma\)-compact)

Proof.

• Suppose that $M$ is a topological manifold. By Lemma 1.10, we can find a countable basis of precompact balls for $M$. Then the closures of these precompact balls constitute a countable compact basis for $M$.

• Suppose that $M$ is Hausdorff, locally Euclidean and $\sigma$-compact.
  Since $M$ is $\sigma$-compact, we can find a countable cover ${(E_i)}_{i\in \mathbb{N}}$ of $M$ consisting of compact sets. Since $M$ is locally Euclidean, for each point $x$ in each compact set $E_i$, there is a local chart $(U_{i,x},{\phi }_{i,x})$. This results in a (possibly uncountable) open cover via the coordinate charts ${(U_{i,x},{\phi }_{i,x})}_{x\in E_i}$ of $E_i$. Since $E_i$’s are compact, we can reduce the open cover ${(U_{i,x},{\phi }_{i,x})}_{x\in E_i}$ to a countable subcover ${(U_{i,j},{\phi }_{i,j})}_{j\in \mathbb{N}}$. This results in an open cover ${(U_{i,j},{\phi }_{i,j})}_{i,j\in \mathbb{N}}$ of $M$.
  For each coordinate chart $(U_{i,j},{\phi }_{i,j})$, the set ${\phi }_{i,j}(U_{i,j})$ is an open set in ${ℝ}^n$. Since ${ℝ}^n$ is second countable, we can find a countable basis ${(B_k)}_{k\in \mathbb{N}}$ of ${\phi }_{i,j}(U_{i,j})$. Since ${\phi }_{i,j}$ is a homeomorphism, the preimages ${\phi }_{i,j}^{-1}(B_k)$, $k\in \mathbb{N}$, constitute a countable basis for $U_{i,j}$. Since the union of local bases is a basis, the collection

  $$\underset{i,j,k\in \mathbb{N}}{\bigcup }\left\{{\phi }_{i,j}^{-1}(B_k)\right\}$$

  is a countable basis for $M$.