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 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 X/~ is homeomorphic to the unit sphere S2.
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.
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