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 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."