Theorem
Given a topological space (X,T) with an equivalence relation R, the quotient set X/R is a topology.
Proof
Define the set of all open sets of
Define a mapping
Then
and
is a topological space so
hence
If
then
and
are open sets in
hence
but 
Hence
is open in
Let
be a family of open sets in
then
and
Hence
is an open subset of
The collection of open sets offorms a topology on