Show that every metrizable space with a countable dense subset has a countable basis.
Show that every metrizable Lindelöf space has a countable basis.
Solution
Goal: Prove that for any metrizable topological space \((X, d)\), separability implies second-countability, and the Lindelöf property implies second-countability.
1. Part (a): Separability implies second-countability for metrizable spaces.Proof:
1.1. Let \(D = \{x_n\}_{n=1}^\infty\) be a countable dense subset of \((X, d)\).
1.2. Define the collection of open metric balls:\[\mathcal{B} = \{B(x_n, r) \mid x_n \in D, \, r \in \mathbb{Q}_+\}.\]
1.3. Since \(D\) and \(\mathbb{Q}_+\) are countable, \(\mathcal{B}\) is a countable collection of open sets in \(X\).
1.4. Let \(U \subseteq X\) be an arbitrary open set, and let \(x \in U\).
1.5. Because \(U\) is open, there is \(\varepsilon > 0\) such that \(B(x, \varepsilon) \subseteq U\).
1.6. Choose a rational radius \(r \in \mathbb{Q}_+\) such that \(r < \varepsilon / 3\).
1.7. Because \(D\) is dense in \(X\), there exists \(x_n \in D\) such that \(d(x, x_n) < r\).
1.8. The ball \(B(x_n, r)\) contains \(x\) because \(d(x, x_n) < r\).
1.9. For any \(y \in B(x_n, r)\), by the triangle inequality:\[d(y, x) \le d(y, x_n) + d(x_n, x) < r + r = 2r < \frac{2\varepsilon}{3} < \varepsilon.\]
1.10. Thus \(x \in B(x_n, r) \subseteq B(x, \varepsilon) \subseteq U\), proving \(\mathcal{B}\) is a countable basis for \(X\).
2. Part (b): Lindelöf property implies second-countability for metrizable spaces.Proof:
2.1. For each integer \(n \ge 1\), the collection of open balls \(\mathcal{U}_n = \{B(x, 1/n) \mid x \in X\}\) covers \(X\).
2.2. Since \(X\) is Lindelöf, there exists a countable subcollection \(\mathcal{C}_n \subseteq \mathcal{U}_n\) covering \(X\), say:\[\mathcal{C}_n = \{B(x_{n, k}, 1/n) \mid k \in \mathbb{Z}_+\}.\]
2.3. Define \(\mathcal{B} = \bigcup_{n=1}^\infty \mathcal{C}_n = \{B(x_{n, k}, 1/n) \mid n, k \in \mathbb{Z}_+\}\).
2.4. As a countable union of countable sets, \(\mathcal{B}\) is a countable collection of open sets.
2.5. Let \(U \subseteq X\) be an open set and \(x \in U\). Choose \(n \ge 1\) such that \(B(x, 2/n) \subseteq U\).
2.6. Since \(\mathcal{C}_n\) covers \(X\), there is some \(k \in \mathbb{Z}_+\) such that \(x \in B(x_{n, k}, 1/n)\).
2.7. For any \(y \in B(x_{n, k}, 1/n)\), the triangle inequality gives:\[d(y, x) \le d(y, x_{n, k}) + d(x_{n, k}, x) < \frac{1}{n} + \frac{1}{n} = \frac{2}{n}.\]
2.8. Thus \(x \in B(x_{n, k}, 1/n) \subseteq B(x, 2/n) \subseteq U\), proving \(\mathcal{B}\) is a countable basis for \(X\).
3. Conclusion:In any metrizable space, separability, Lindelöfness, and second-countability are all equivalent. Q.E.D.