Theorem
A set with the order topology is a T2 space.
Proof
Suppose
is an ordered set and let
represent the family of subsets of
of the form
or
for
The familyis a subbasis for a topology
on
called the order topology oninduced by
Let
represent distinct points. Sinceis ordered, either
or
Suppose then there are two possibilities.
1. An element
exists such that
Then
and
are disjoint neighbourhoods of
and
respectively.
2. Noexists such that
thenandare disjoint neighbourhoods ofandrespectively.