How does one show a topological space is metrizable? Using text Intro. to Topo. by Mendelson I'm self studying Intro. to Topology by Mendelson.
The problem I'm looking at is,
Prove that for each set $X$, the topological space $(X,2^X)$ is metrizable.
I'm not having so much trouble with this problem per se, but with the idea of metrizable spaces. From what I've understood from the reading is that a metric space $(X,d)$ satisfies a theorem which is exactly the definition of a topological space using only open sets from $X$ and so these metric spaces are known as metrizable spaces. Is my thought process sound?
Going back to this problem, would I need to find a metric $d$ that can induce/create the topology on $X$? Would I need to look at the open sets of $X$? Am I even approaching this problem correctly?
Thanks for any feedback or hints.
 A: It is better to think of topological metrizable spaces as "metric spaces with the metric forgotten". You see, a metric space has an additional structure, namely the metric, but also "happen" to induce a topology via its open sets. So, the reason for having the two names, "metrizable space" and "metric space", is to differentiate between what we're looking at: If we say "a metric space", we mean a set with a metric function which defines distance betweeen points, satisfies the triangle inequality and so forth. If we say "a metrizable space", We mean a topological space, with open sets as its entire structure, but we also know that this topological space came from a metric, and hence satisfies many "good" qualities such as separation axioms.
Metric spaces are trivially topological spaces, but proving that a topological space is metrizable is generally hard. One way to do it is "recall" the metric. If you could realize what the metric that induced the topological space was, and show that the topology induced by it is exactly the topology of the given topological space, then you've shown the space is metrizable.
I was going to let you figure out what the metric is in your case, but Aneesh already told you.
