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 makes
a T1 space is the discrete topology.
Proof
Letrepresent a finite set and let
be a topology on
such that
is a T1 space.
Sinceis a T1 space every singleton set
is closed. Each finite union of closed sets is closed, and since
is a finite set, all unions of sunsets of
are closed. Hence, all subsets of
are open and
is the discrete topology.