Proof That Every Metric Space is First Countable
Every metric space is a first countable space.
A topological spaceis first countable if, for eacha countable setof open sets, each set containingexists with each open setcontainingcontains a point ofThat is,is first countable if and only if, at every pointa countable local base exists.
Letbe a metric space and letThe set of open balls
forms a countable local basis at
For every open setcontaininga ballexists.
Hence every metric space is a first countable space.