Quotient map
From Topospaces
This article defines a property of continuous maps between topological spaces
Contents
Definition
Symbol-free definition
A continuous map between topological spaces is termed a quotient map if it is surjective, and if a set in the range space is open iff its inverse image is open in the domain space.
Definition with symbols
Let be topological spaces and
be continuous maps.
is termed a quotient map if it is sujective and if
is open iff
is open in
.
Significance
Given a topological space , a set
and a surjective map
, we can prescribe a unique topology on
, the so-called quotient topology, such that
is a quotient map. Moreover, this is the coarsest topology for which
becomes continuous.
Also, the study of a quotient map is equivalent to the study of the equivalence relation on given by
.