X is a set and P(X) is the discrete topology on X, meaning that P(X) consists of all subsets of X. I want to prove that X is metrizable.
My text says that a topological space X is metrizable if it arises from a metric space. This seems a little unclear to me, which is probably why I am slightly confused.
The Attempt at a Solution
This problem seems really easy, I am just unsure of what I am supposed to prove. I want to show that X, together with the discrete topology, is metrizable. I choose the discrete metric d, which is defined by d(x,y) = 0 if x=y and d(x,y) = 1 if x=/=y.
This is where I am unsure of what I am supposed to show. I can start by showing that if we have the metric space (X,d), then every subset of X is open since all the points are isolated. Then P(X) satisfies the property of a topology on X, so (X,P(X)) is a topological space. But I don't think this is correct because we already assumed (X,P(X)) is a topological space. In fact, it is ALWAYS a topological space for any X, right?
Am I supposed to show that if (X,P(X)) is a topological space, (X,d) is a metric space? But this is also obvious because I already know that d is a metric.
What am I supposed to be proving?