 First-countable space

A topological space is first-countable when each point has a countable neighbourhood basis. Every metric space is first-countable: the open balls $B(x,1/n)$ form such a basis. A subset of a first-countable space remains first-countable in its subspace topology, since intersecting a local basis with the subset preserves the defining containment property. Thus a space failing first countability cannot be homeomorphic to any subset of a metric space.

In topology, a topological space \( X \) is called *first-countable* if, at every point \( x \in X \), there exists a countable collection of open sets (called a *countable neighborhood basis*) such that any open neighborhood of \( x \) contains at least one of these open sets.