Prove that every manifold is regular and hence metrizable. Where do you use the Hausdorff condition?
Solution
1. Definition of topological manifold:
1.1. An \(m\)-dimensional topological manifold \(M\) is a Hausdorff space with a countable basis that is locally Euclidean of dimension \(m\) (every point \(x \in M\) has an open neighborhood homeomorphic to an open ball in \(\mathbb{R}^m\)).
2. Proof that \(M\) is regular:
2.1. A \(T_1\) space is regular if and only if for every point \(x \in M\) and every open neighborhood \(V\) of \(x\), there exists an open set \(W\) containing \(x\) such that \(\overline{W} \subseteq V\).
2.2. Let \(x \in M\) and let \(V\) be an open neighborhood of \(x\).Since \(M\) is locally Euclidean, there is an open neighborhood \(U\) of \(x\) and a homeomorphism \(\phi: U \to B(0, 1) \subset \mathbb{R}^m\) such that \(\phi(x) = 0\).The set \(\phi(U \cap V)\) is open in \(\mathbb{R}^m\) and contains \(0\).
2.3. Choose \(r \in (0, 1)\) sufficiently small so that the closed Euclidean ball \(\overline{B(0, r)} \subset \phi(U \cap V)\).Let \(W = \phi^{-1}(B(0, r))\) and \(K = \phi^{-1}(\overline{B(0, r)})\).Since \(\overline{B(0, r)}\) is compact in \(\mathbb{R}^m\) and \(\phi^{-1}\) is continuous, \(K\) is a compact subset of \(M\).
2.4. Role of the Hausdorff condition: Because \(M\) is Hausdorff, every compact subset of \(M\) is closed in \(M\).Therefore \(K\) is closed in \(M\).Since \(W \subseteq K\) and \(K\) is closed, the closure of \(W\) in \(M\) satisfies:\[ \overline{W} \subseteq K \subseteq U \cap V \subseteq V. \]Thus \(M\) is regular.
3. Proof of metrizability via Urysohn’s Metrization Theorem:
3.1. By definition, a manifold \(M\) has a countable basis (second-countable).By 2, \(M\) is regular.
3.2. By the Urysohn Metrization Theorem, every regular, second-countable space is metrizable.Therefore \(M\) is metrizable.
4. Discussion of where Hausdorff is used:
4.1. The Hausdorff condition was used crucially in 4 to ensure that the compact set \(K = \phi^{-1}(\overline{B(0, r)})\) is closed in \(M\), ensuring \(\overline{W} \subseteq K \subseteq V\).Without the Hausdorff condition, locally Euclidean second-countable spaces (such as the line with two origins) fail to be Hausdorff, fail to be regular (the two origins cannot be separated), and are not metrizable.
5. Conclusion:Manifolds are regular and metrizable, with the Hausdorff condition guaranteeing that compact Euclidean neighborhoods are closed. Q.E.D.