So, my professor asked me to prove a theorem, and I only have a question about PART of it.
Theorem: Let be a set and let be a metric on . The set forms a basis for a topology on .
My question is this: Is it necessary to union the empty set with the open balls? I don't understand why this is necessary.
Actually, the only instance that I can think of where you would need the empty set as part of the basis would be if the set consisted of only one point. So long as the set contains at least two points, metrizability will ensure that there exists two disjoint non-empty open sets, and the empty set would not be necessary to create either (either through unions or finite intersections). So, any basis for a space with at least two points will pick up the empty set through finite intersections of disjoint open sets. Seems to me that the single point set is the only exception. (Unless you consider the empty set to be metrizable...)