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