Proof That Every Metric Space is a T1 Space

Theorem

Every Metric Space is a T1 space.

Proof

A topological spaceis called a T1 space if every singleton setis closed.

In a metric spacethe condition for a setto be closed can be written aswith eachso that for each sequence tending to a limit point, the limit point is in

We can define a sequenceand a setObviouslyso that every metric space is a T1 space.

A topological space containing two points with the indiscrete topology is not a T1 space. If tandthenis not a T1 space.

Every T1 space is a T0 space, since if a singleton setin a spaceis closed andis any other point ofthenis open and