## Proof That Any Two Points of a T1 Space Are in Open Sets Not Containing the Other Point

Theorem

If a topological spaceis a T1 space then for any pair of distinct pointsopen setsexist with

Proof

Supposeis a T1 space. For anyis closed. Letbe distinct points ofandare closed soandare open.

Alsoandand

Conversely supposeis open. Letthenand an open setexists such thatHenceand

Since allare openis open andis closed so