Proof That a Topology Determined by a Basis is Unique

Theorem

Ifis a basis of a topologythen the topologyis unique.

Proof

Letrepresent any set and letrepresent a collection of subsets ofSuppose is a basis for distinct topologiesand

Sinceandare distinct, at least one subsetexists such thatbutor vice versa. Since alsois a basis forandwhere

Sinceis a basis forany union of elements ofmust be a member of

This is a contradiction sinceandare distinct, henceand a basis determines a unique topology.