Proof That a Topological Space is Regular if and Only Every Element is in an Open Set That Has a Closed Subset Containing the Element

Theorem

A spaceis regular if and only if for everyand every open neighbourhoodof contains a closed neighbourhoodof with

Proof

Supposeis regular and letbe a point ofLetbe an open neighbourhood ofthenis closed and

Hence open setsandexist such that

Sincewe haveSincewe have

Hence

Now supposeandis an open subset ofwithAn open neighbourhoodofexists with

Letand letbe any closed subset ofwithis an open neighbourhood ofthen an open setexists withand

is open and

so

Henceandare the required open sets andis regular.