Show that any countable space with the discrete topology is separable.
Solution
1. Definition of separability:
1.1. A topological space \((X, \mathcal{T})\) is separable if there exists a subset \(D \subseteq X\) such that:(i) \(D\) is at most countable (\(|D| \le \aleph_0\)), and(ii) \(D\) is dense in \(X\) (\(\overline{D} = X\)).
standard definition of separability.
2. Verification for a countable discrete space:
2.1. Let \(X\) be a countable space equipped with the discrete topology \(\mathcal{T} = \mathcal{P}(X)\).Choose the subset \(D = X \subseteq X\).
choice of \(D\).
2.2. \(D = X\) is countable by the hypothesis that \(X\) is countable.
hypothesis.
2.3. The closure of the whole space is \(\overline{D} = \overline{X} = X\), so \(D\) is dense in \(X\).
property of topological closure.
3. Conclusion:Since \(D = X\) is a countable dense subset of \(X\), \((X, \mathcal{T})\) is separable. Q.E.D.
1 and 2.