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Proof That a Set With the Order Topology is a T2 Space

Theorem

A set with the order topology is a T2 space.

Proof

Supposeis an ordered set and letrepresent the family of subsets ofof the formorfor

The familyis a subbasis for a topologyoncalled the order topology oninduced by

Letrepresent distinct points. Sinceis ordered, eitherorSuppose then there are two possibilities.

1. An elementexists such that

Thenandare disjoint neighbourhoods ofandrespectively.

2. Noexists such thatthenandare disjoint neighbourhoods ofandrespectively.