I guess that by "equivalent" you meant homeomorphic. in the discrete space every point is an open set, meaning is an open set.

now, for where A is finite (nonempty) subspace, define , or m=1 if |A|=1. because A is finite, m is well defined. now for every the ball of radius m/2 around x contain only x, so {x} is an open set - therefore this topology is homeomorphic to the discrete topology