Quotient Space
The quotient space 
of a topological space 
and an equivalence relation 
on 
is the set of equivalence classes of points in 
(under the equivalence relation 
) together with the following topology given to subsets of 
: a subset 
of 
is called open iff ![union _([a] in U)a](https://mathworld.wolfram.com/images/equations/QuotientSpace/Inline10.gif)
is open in 
. Quotient spaces are also called factor spaces.
This can be stated in terms of maps as follows: if 
denotes the map that sends each point to its equivalence class in 
, the topology on 
can be specified by prescribing that a subset of 
is open iff ![q^(-1)[the set]](https://mathworld.wolfram.com/images/equations/QuotientSpace/Inline16.gif)
is open.
In general, quotient spaces are not well behaved, and little is known about them. However, it is known that any compact metrizable space is a quotient of the Cantor set, any compact connected 
-dimensional manifold for 
is a quotient of any other, and a function out of a quotient space 
is continuous iff the function 
is continuous.
Let 
be the closed 
-dimensional disk and 
its boundary, the 
-dimensional sphere. Then 
(which is homeomorphic to 
), provides an example of a quotient space. Here, 
is interpreted as the space obtained when the boundary of the 
-disk is collapsed to a point, and is formally the "quotient space by the equivalence relation generated by the relations that all points in 
are equivalent."
REFERENCES:
Munkres, J. R. Topology: A First Course, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2000.
