Lee Problem 1.3

First, suppose that \(X\) is locally Euclidean Hausdorff. Assume that \(X\) is \(\sigma\) -compact, so \(X = X_1 \cup X_2 \cup \cdots\), where each \(X_k\) is compact. For some \(x \in X\), \(x \in U_x\), the domain of some coordinate chart \(\varphi : U_x \rightarrow \hat{U}_x \subset \mathbb{R}^n\). We take a countable basis for each \(\hat{U}_x\), which we call \(\mathcal{B}_x\). The countable collection of open sets \(\varphi^{-1}(\mathcal{B}_x)\) is clearly a countable basis for the subspace \(U_x\).

Since each \(X_k\) is compact, each is covered by a finite number of the \(U_x\), so their countable union is covered by a countable number of the \(U_x\). We combine the corresponding countable collection of countable bases, to get the countable set \(\mathcal{S}\) of sets open in \(X\). Taking all finite intersections of elements of \(\mathcal{S}\) will result in a new countable set \(\mathcal{S}'\): we claim that this is a basis for \(X\). Indeed, given \(U\) open in \(X\), and \(y \in U\), \(y\) will be in some \(U_x\) in the countable cover. \(U \cap U_x\) is open in \(U_x\), so we can find \(y \in \varphi^{-1}(B_x) \subset U_x\) for some \(B_x \in \mathcal{B}_x\).

Conversely, suppose that \(X\) is locally Euclidean Hausdorff and second-countable (i.e. a topological manifold). We know from Lemma 1.10 that it has a countable basis of precompact coordinate balls, so we simply take the closure of these balls to get a countable collection of compact sets whose union in \(X\). Thus, \(X\) is \(\sigma\) -compact by definition.