7.9 First- and second-countable spaces[2BK]
[0MC] A topological space satisfies the first axiom of countability if each point admits a fundamental system of neighborhoods that is countable.
[0MD]A topological space satisfies the second axiom of countability when it has a countable base.
- E267
[0MF] Difficulty:*.If \((X,𝜏)\) satisfies the second axiom of countability, if \(A⊆ X\) is composed only of isolated points, then \(A\) has countable cardinality. Hidden solution: [UNACCESSIBLE UUID ’0MG’]
- E267
[0MH] Prerequisites:2. If \((X,𝜏)\) satisfies the second axiom of countability, given \(A⊆ X\) there exists a countable subset \(B⊆ A\) such that \(\overline B⊇ A\). In particular, the whole space \(X\) admits a dense countable subset: \(X\) is said to be separable. The vice versa holds for example in metric spaces, see 285. See also 3 for an application in \(ℝ^ n\).
Hidden solution: [UNACCESSIBLE UUID ’0MJ’]
The countability axioms will return in exercises 285 and 285.