First countable space: a countable neighbourhood base at every point

A topological space $(X, \mathcal{T})$ (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison) is first countable if every point of $X$ has an at most countable neighbourhood base: for each $x \in X$ there is a family $\mathcal{B}_x \subseteq \mathcal{N}(x)$ that is at most countable (Finite, countably infinite, countable, uncountable, Equinumerous sets, $A \approx B$ and $A \preceq B$) and such that every neighbourhood of $x$ contains a member of $\mathcal{B}_x$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

"Countable" here means "at most countable", as everywhere in this library (Finite, countably infinite, countable, uncountable), so a finite neighbourhood base is permitted. That is not a degenerate case: in a discrete space the one-element family $\{\{x\}\}$ is a neighbourhood base at $x$, so every discrete space is first countable, and in an indiscrete space $\{X\}$ is a neighbourhood base at every point.

The base may be taken to consist of open sets, and it may be taken decreasing. If $\mathcal{B}_x$ is an at most countable neighbourhood base at $x$, then replacing each $N \in \mathcal{B}_x$ by an open $U_N$ with $x \in U_N \subseteq N$ gives an at most countable neighbourhood base of open sets. Making the base decreasing, that is arranging $M_0 \supseteq M_1 \supseteq \dots$, requires enumerating it and forming the running finite intersections; both operations are carried out inside the proof of the theorem that uses them, the next item, where the enumeration and the recursion are cited explicitly rather than assumed here.

First countability is a topological property (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological): a homeomorphism $h : X \to Y$ carries a neighbourhood base at $x$ to a neighbourhood base at $h(x)$, since $N \mapsto h[N]$ is a bijection between the neighbourhood filters preserving inclusion, and a bijection preserves at most countability (Equinumerous sets, $A \approx B$ and $A \preceq B$).

• What first countability buys. Under countable choice, it is a sufficient hypothesis under which sequences detect the topology: the closure is the sequential closure and sequential continuity is continuity. It is not necessary: the later hierarchy Assuming countable choice, every first countable space is Fréchet–Urysohn; in ZF every Fréchet–Urysohn space is sequential ↗ distinguishes first countable, Fréchet--Urysohn, and sequential spaces. Without an additional hypothesis sequences can be too weak. Under that same choice assumption, both failures occur in the cocountable topology on $\mathbb{R}$, so the cited implication shows that space is not first countable.

• Every metric space is first countable, the balls of radius $1/n$ for $n \ge 1$ forming an at most countable neighbourhood base at each point (The balls $B(x, 1/n)$, $n \ge 1$, form a countable neighbourhood base at $x$, so every metric space is first countable); so every metrizable space is first countable, and a space that is not first countable is not metrizable. That is a second obstruction to metrizability, and it is not stronger than the Hausdorff obstruction used elsewhere on this page: the indiscrete topology on two points is first countable, as the paragraph above records, and is not Hausdorff, so it is caught by the Hausdorff obstruction and not by this one. The converse failure does occur under the ultrafilter lemma and countable choice: with those hypotheses, the later Cantor-cube example Assuming the Ultrafilter Lemma and Countable Choice, an uncountable Cantor cube is compact Hausdorff and uniformizable but not first countable, hence not metrizable ↗ is compact Hausdorff and not first countable. Together with the indiscrete example, this shows under those hypotheses that neither first countability nor Hausdorffness implies the other.

• Second countability is not developed at this point in the reading order. The stronger axiom, an at most countable basis for the whole topology, is defined later in Second countability: an at most countable basis for the topology ↗. Nothing below uses it.