Proof That the Quotient Topology is a Topology
Given a topological space (X,T) with an equivalence relation R, the quotient set X/R is a topology.
Define the set of all open sets of
Define a mapping
is a topological space sohence
Ifthenandare open sets inhencebut Henceis open in
Letbe a family of open sets inthenandHenceis an open subset of
The collection of open sets offorms a topology on