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Quotient space

In topology, a quotient space is (intuitively speaking) the result of identifying or "gluing together" certain points of some other space. The points to be identified are specified by an equivalence relation. This is commonly done in order to construct new spaces from given ones.

Formally, suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/em> is open if and only if their union is open in X.

Examples

Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y iff x-y is an integer. Then the quotient space X/~ (also written as R/Z) is homeomorphic to the unit circle S1.

As another example, consider the unit square X = [0,1]×[0,1] and the equivalence relation ~ generated by the requirement that all boundary points be equivalent, thus identifying all boundary points to a single equivalence class. Then is homeomorphic to the unit sphere S2.

Properties

Let p : X be the projection map which sends each element of X is continuous; in fact, the topology on is the finest (the one with the most open sets) which makes p is in general not open.

If Y is some other topological space, then a function f : Yop is continuous.

If g : XY is a continuous map with the property that a~b implies g(a)=g(b), then there exists a unique continuous map h : Y = hop.

The continuous maps defined on are therefore precisely those maps which arise from continuous maps defined on X