firstcountable

First-countable, or first-countable spaces, is a property of a topological space X described by a local base condition: for every point x in X there exists a countable collection {U_n} of neighborhoods of x such that every neighborhood of x contains some U_n.

Equivalently, a space is first-countable if each point has a countable neighborhood base. This condition has important consequences for convergence: in a first-countable space, a set A is closed if and only if whenever a sequence in A converges to a point x in X, the limit x is in A. Thus, first-countable spaces are sequential, meaning that closure coincides with sequential closure.

Many familiar spaces are first-countable. Any metric space is first-countable, since metric balls around a point form a countable base of neighborhoods. The Sorgenfrey line is another example: its neighborhoods can be generated by a countable family at each point, so it is first-countable, even though it is not second-countable. In general, second-countable spaces are first-countable, but the converse need not hold.

Subspaces of first-countable spaces inherit the property: if X is first-countable and Y ⊆ X with the subspace topology, then Y is first-countable as well.

Products present a contrast: the product of countably many first-countable spaces is first-countable, but the product of uncountably many nontrivial spaces need not be. Understanding first-countability helps clarify how local structure affects convergence and decidability of closure in topological spaces.