- What is a separable space?
Solution
1. Definition of a separable space:
1.1. A topological space \((X, \mathcal{T})\) is called separable if there exists a subset \(D \subseteq X\) such that:1. \(D\) is at most countable (i.e., \(|D| \le \aleph_0\)),2. \(D\) is dense in \(X\), meaning that the topological closure of \(D\) equals \(X\):\[ \overline{D} = X. \]
2. Equivalent characterizations:
2.1. A space \(X\) is separable if and only if there exists a countable set \(D \subseteq X\) that intersects every non-empty open set:\[ \forall U \in \mathcal{T} \setminus \{\emptyset\}, \qquad U \cap D \neq \emptyset. \]
3. Properties and examples:
3.1. Every second-countable space is separable (choosing one point from each element of a countable basis).
3.2. For metric spaces, separability is equivalent to being second-countable and equivalent to being Lindelöf.
3.3. The Euclidean space \(\mathbb{R}^n\) with the standard topology is separable, with countable dense subset \(\mathbb{Q}^n\).
4. Conclusion:A separable space is a topological space containing a countable dense subset. Q.E.D.