The Banach Category Theorem

Abstract

In a topological space that has a countable base, it is obvious that the union of any family of open sets of first category is of first category. One need only take the union of those members of the base that are contained in at least one member of the given family. The same reasoning shoves that the union of any family of open sets of measure zero has measure zero (for any measure defined for all open sets). It is remarkable that the first statement remains valid whether the space has a countable base or not. The second statement, however, needs to be qualified.