Let we need to prove that for any there exists such that if . Pick then for to be true we require that since it is the discrete metric. But then clearly .

A function from metric spaces and is continous if and only if the inverse-image of every open set is open. Since is a trivial topology it means the only open sets are and . But and . But and are open. Thus, was a continous function.Let y have the trivial topology and X be an arbitrary topological space. Show that every funtion f: X map to Y is coninuous.

Let be a closed subset of . Consider (the complement). This set is open. And since is continous it means is open. Thus, is closed. The converse is similar.Let X & Y be topological spaces. A function f:X maps to Y is continuous if and only in the inverse (C) is closed in X for every closed set C is a subset of Y.