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 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 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."