X T SolutionAny subset R X X is said to be a relation in X

Solution

Any subset R X × X is said to be a relation in X. We write x R y if (x, y) R. An equivalence relation in a set X is a relation R in X such that the following conditions are satisfied.

Let X be a discrete topological space, and let be an equivalence relation on X. Let p: X X/ be the quotient map. By definition of quotient topology, a subset U of X/ is open if and only if p 1 (U) is an open subset of X. But every subset of X is open (since X has the discrete topology). Hence, every subset of X/ is open; that is to say, X/ is discrete.