quotient space

Let be a topological space^{👁 Mathworld
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, and let be an equivalence relation^{👁 Mathworld
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on . Write for the set of equivalence classes^{👁 Mathworld
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of under . The quotient topology on is the topology^{👁 Mathworld
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whose open sets are the subsets such that

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

The projection map which sends each element of to its equivalence class is always a continuous map^{👁 Mathworld
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. In fact, the map satisfies the stronger property that a subset of is open if and only if the subset of is open. In general, any surjective map that satisfies this stronger property is called a quotient map, and given such a quotient map, the space is always homeomorphic to the quotient space of 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 is continuous^{👁 Mathworld
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.

For , one often writes for the quotient space obtained by identifying all the points of 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