Definition
Let (X,τX) be a topological space, and let ~ be an equivalence relation on X. The quotient space, is defined to be the set of equivalence classes of elements of X:
equipped with the topology where the open sets are defined to be those sets of equivalence classes whose unions are open sets in X:
Equivalently, we can define them to be those sets with an open preimage under the quotient map which sends a point in X to the equivalence class containing it.
The quotient topology is the final topology on the quotient space with respect to the quotient map.
Read more about this topic: Quotient Space
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