Proof That the Only Topology on a Finite Set X Which Makes X a T1 Space is the Discrete Topology

Theorem

Ifis a finite set, the only topology which makesa T1 space is the discrete topology.

Proof

Letrepresent a finite set and letbe a topology onsuch thatis a T1 space.

Sinceis a T1 space every singleton setis closed. Each finite union of closed sets is closed, and sinceis a finite set, all unions of sunsets ofare closed. Hence, all subsets of are open andis the discrete topology.