Proof That a Two Element Space With the Indiscrete Topology is Not a T0 Space
Letwith the indiscrete topology is not a T0 space.
A spaceis a T0 space if for any two distinct elementsthere is an open neighbourhood of one which does not contain the other.
With the indiscrete topologywe cannot separate 0 from 1 or 1 from 0 henceis not a T0 space.
Consider the setwith the topologyFor the distinct points 0 and 1, an open setexists such thatbut henceis a T0 space.
Each metric space is a T0 space since for distinct pointsandthere existssuch thathencebut