Proof That a Topology T is a Discrete Topology on a Set X if and Only if Every Point of X is an Open Set

Theorem

A Topologyis a Discrete Topology on a Setif and Only if Every Point ofis an Open Set.

Proof

Letthe set of all collections of elements ofis called the discrete topology onEvery point ofis an open set.

Sinceconsists of all subsets of

Conversely, supposeis a topology onsuch that

Letbe any subset ofthen

All setsare open sets ofsoand sinceis a topology, the union of open sets is an open set. Hence, sincewe have