Topology/Quotient Spaces

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Let (X,τ) be a topological space, and let f be a function from X to a set Y.

We define the quotient space to be the topology on Y such that a set S in Y is open when its preimage under the function f, f ^{− 1}(S), is also open.

Example: let X be the real numbers with the usual topology, and let Y be the set following equivalence classes of the real numbers with the equivalence relation defined as follows:

x~y if x-y is an integer

Which is obviously an equivalence relation.

Now let the function f be defined as follows:

f(x)=the equivalence class containing x.

It can easily be proved that this quotient space is homeomorphic to a circle.