## Proof That a Completely Regular Space is Regular

Theorem

A completely regular space is regular.

Proof

A topological spaceis completely regular if, for any closed subsetofand anya continuous functionexists such that for every and

By hypothesis a continuous functionexists such thatandThe intervalis Hausdorff. Hence open disjoint subsetsandofexists such thatand sinceis continuousandare open andand

Henceis regular.