## Proof That the Quotient Topology is a Topology

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

Thenand

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