I'm looking at Munkres' Topology (2nd edition), trying to refresh myself on some of the basics. I'm having an issue with an early Lemma of his (Lemma 13.1), which states:
Let X be a set, and let B be a basis for a topology T on X. Then T equals the collection of all unions of elements of B.
My issue with this is that any topology contains the empty set, but nothing about the definition* of a basis for a topology requires the empty set to be in B. Indeed, Munkres' proof of this Lemma ignores the empty set in T altogether. For this being such a well reputed book, I'm wondering if I'm missing something. Any thoughts?
*Definition of a basis in Munkres:
If X is a set, a basis for a topology on X is a collection B of subsets of X such that:
-For each x in X, there is at least one basis element U in B containing X.
-If x belongs to the intersection of two basis elements U and V, then there is a basis element W containing x such that W is contained in the intersection of U and V.