firstcountableness

First-countability is a property of a topological space that says every point has a countable local base. More precisely, a space X is first-countable if for every x in X there exists a sequence of neighborhoods U1 ⊆ U2 ⊆ ... of x such that every neighborhood of x contains some Un. This sequence is called a local base at x. If this holds for all points x, the whole space is called first-countable.

All metric spaces are first-countable: at a point x you can take the shrinking balls B(x,1/n) for n = 1, 2, 3, ..., which form a countable neighborhood basis. The property is weaker than second-countability; there exist spaces that are first-countable but not second-countable. Classic examples include the long line and the Sorgenfrey line, both first-countable yet not second-countable and not metrizable.

Subspaces of first-countable spaces inherit the property: if X is first-countable and Y ⊆ X, then Y is first-countable in the subspace topology. Finite products of first-countable spaces are first-countable; however, arbitrary (in particular uncountable) products need not be.

A key consequence is that first-countable spaces are sequential: in such spaces, a set is closed if and only if it contains the limits of all sequences from that set. Equivalently, sequence convergence suffices to describe closure, which is not true in all topological spaces.

In summary, first-countability ensures manageable local structure around each point through sequences, is enjoyed by all metric spaces, but is not in general enough to guarantee metrizability or second-countability.