Formally, suppose X is a topological space and ~ is an equivalence relation on X. We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X.
Consider the set X = R of all real numbers with the ordinary topology, and write x ~ y iff xy is an integer. Then the quotient space X/~ (also written as R/Z) is homeomorphic to the unit circle S^{1}.
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 X/~ is homeomorphic to the unit sphere S^{2}.
Let p : X → X/~ be the projection map which sends each element of X to its equivalence class. The map p is continuous; in fact, the topology on X/~ is the finest (the one with the most open sets) which makes p continuous. The map p is in general not open[?].
If Y is some other topological space, then a function f : X/~ → Y is continuous if and only if fop is continuous.
If g : X → Y is a continuous map with the property that a~b implies g(a)=g(b), then there exists a unique continuous map h : X/~ → Y such that g = hop.
The continuous maps defined on X/~ are therefore precisely those maps which arise from continuous maps defined on X that respect the equivalence relation (in the sense that they send equivalent elements to the same image). This criterion is constantly being used when studying quotient spaces.
Compatibility with other topological notions
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