1. The problem statement, all variables and given/known data If A is a subspace of X and A has discrete topology does X have discrete topolgy?Also if X has discrete topology then does it imply that A must have discrete topology? 3. The attempt at a solution My understanding of discrete topology suggests to me that if A is discrete it doesn't imply that X is also. example A the natural numbers as a subspace of X the real numbers with the euclidean metric. I feel the reverse is true. For X to be discrete does every singleton in the metric space have to be open? ie: is it correct to say that the a metric topology is discrete if and only if each singleton in the metric space is open? I think the answer is yes in which case each singlton in X is open meaning any subspace is also discrete. Since the subspace would be a collection of singletons and unions of singletons. Also for a set to be open is true to say: a set A a subset of X is open if and only if for all a in A d(a,X-A)>0?