Proof That a Topology Determined by a Basis is Unique
Theorem
If
is a basis of a topology
then the topologyis unique.
Proof
Let
represent any set and letrepresent a collection of subsets of
Suppose is a basis for distinct topologies
and
Sinceand
are distinct, at least one subset
exists such that
but
or vice versa. Since also
is a basis forand
where 
Sinceis a basis forany union of elements ofmust be a member of
This is a contradiction sinceandare distinct, hence
and a basis determines a unique topology.
