Countability axioms control how much local or global topological information can be encoded by countable data. They are the reason sequences and countable constructions work in many spaces.
First countability
First countability gives each point a countable neighborhood scale.
Definition. A space is first countable if every point \(x\) has a countable local basis \(\{B_n(x)\}_{n\ge1}\), meaning every neighborhood of \(x\) contains some \(B_n(x)\).
Proposition. In a first-countable space, \(x\in\overline A\) if and only if there is a sequence in \(A\) converging to \(x\).
Proof. If a sequence from \(A\) converges to \(x\), every neighborhood of \(x\) meets \(A\), so \(x\in\overline A\). Conversely, let \(\{B_n\}\) be a countable local basis and replace it by the decreasing basis \(C_n=B_1\cap\cdots\cap B_n\). Since \(x\in\overline A\), choose \(a_n\in A\cap C_n\). Then \(a_n\to x\).
Remark. Without first countability, closure may require nets rather than sequences.
Second countability
Second countability gives the whole space a countable basis.
Definition. A space is second countable if its topology has a countable basis.
Proposition. Every second-countable space is first countable and Lindelöf.
Proof. For first countability, collect all basis elements containing a fixed point. For Lindelöfness, let \(\mathcal U\) be an open cover and \(\mathcal B=\{B_n\}\) a countable basis. For each basis element \(B_n\) contained in some member of \(\mathcal U\), choose one such member. The chosen countable subfamily covers every point because each point lies in some cover element and hence in a basic set contained in it.
Separability in metric spaces
In metric spaces, countable dense sets and countable bases are equivalent.
Definition. A space is separable if it has a countable dense subset.
Theorem. For metric spaces, second countability and separability are equivalent.
Proof. If a metric space has a countable basis, choose one point from each nonempty basis element; the chosen set is countable and dense. Conversely, if \(D\) is countable and dense, the balls \(B(d,q)\) with \(d\in D\) and rational \(q>0\) form a countable basis. Given \(x\in U\) open, choose \(r>0\) with \(B(x,r)\subseteq U\), then choose \(d\in D\cap B(x,r/3)\) and rational \(q\) with \(r/3<q<2r/3\); \(x\in B(d,q)\subseteq U\).
Lindelöf spaces
The Lindelöf property is a countable compactness-like extraction principle for open covers.
Definition. A space is Lindelöf if every open cover has a countable subcover.
Proposition. A closed subspace of a Lindelöf space is Lindelöf, and a continuous image of a Lindelöf space is Lindelöf.
Proof. For a closed subspace \(F\), add \(X\setminus F\) to a cover of \(F\) and take a countable subcover of \(X\). For continuous images, pull an open cover of \(f(X)\) back to \(X\), choose countably many pullbacks covering \(X\), and push the corresponding cover elements forward.
Example. The Sorgenfrey line is separable and first countable but not second countable; products of Lindelöf spaces need not be Lindelöf.