quotient space

Let X be a topological spaceMathworldPlanetmath, and let be an equivalence relationMathworldPlanetmath on X. Write X* for the set of equivalence classesMathworldPlanetmath of X under . The quotient topology on X* is the topologyMathworldPlanetmath whose open sets are the subsets UX* such that


is an open subset of X. The space X* is called the quotient space of the space X with respect to . It is often written X/.

The projection map π:XX* which sends each element of X to its equivalence class is always a continuous mapMathworldPlanetmath. In fact, the map π satisfies the stronger property that a subset U of X* is open if and only if the subset π-1(U) of X is open. In general, any surjective map p:XY that satisfies this stronger property is called a quotient map, and given such a quotient map, the space Y is always homeomorphic to the quotient space of X under the equivalence relation


As a set, the construction of a quotient space collapses each of the equivalence classes of to a single point. The topology on the quotient space is then chosen to be the strongest topology such that the projection map π is continuousMathworldPlanetmath.

For AX, one often writes X/A for the quotient space obtained by identifying all the points of A with each other.

Title quotient space
Canonical name QuotientSpace
Date of creation 2013-03-22 12:39:40
Last modified on 2013-03-22 12:39:40
Owner djao (24)
Last modified by djao (24)
Numerical id 5
Author djao (24)
Entry type Definition
Classification msc 54B15
Related topic AdjunctionSpace
Defines quotient topology
Defines quotient map