The quotient space 
of a topological space 
and an equivalence relation 
on 
is the set of equivalence
classes of points in 
, equipped with the quotient topology.
Quotient spaces are also called factor spaces.
Let 
be the map sending each point to
its equivalence class. The topology
on 
is defined by declaring a subset 
to be an open
set iff its inverse image 
is an open
set in 
.
A function 
to a topological
space 
is continuous iff 
is continuous.
Every nonempty compact space with a metrizable topology is a quotient of the Cantor set. Every
nonempty compact connected 
-dimensional manifold, for 
, is a quotient of every other such 
-dimensional manifold.
Let 
be the 
-dimensional
closed disk, where 
is an integer, and let 
be its boundary,
the 
-dimensional
hypersphere. Then 
, obtained by collapsing the boundary
of the closed disk to a point,
is homeomorphic to the hypersphere 
. Here the equivalence
relation identifies all points in 
and leaves each other point
in its own equivalence class.
