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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“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
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—François, Duc De La Rochefoucauld (1613–1680)