Hello,
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.
don't get lost in the formalism. a base for a topology is just something we can recover the topology by "unioning arbitrary elements of". for example, a base for the standard topology on the real line is the set of all open intervals B = {(a,b): a,b in R}. note that this example includes the empty interval (a,a). you can think of a base as being sort of a generating set for a topology.
note that if we speak of "the topology generated by (a base) B", then whether or not B includes the empty set is largely irrelevant, as every topology includes the empty set.
if B is closed under (finite) intersections, it's clear we automatically get a topology. but B need not be closed under intersections. if not, we have, for any U_{1},U_{2} in B:
U_{1} ∩ U_{2} = U {U_{x} in B : U_{x} ⊂ U_{1} ∩ U_{2}, x in U_{1} ∩ U_{2}},
and the definition of a base guarantees the existence of such U_{x}, for every x in the intersection of U_{1} and U_{2}, which shows that the intersection of any two elements of B is contained in the set of unions of elements of B.
if U_{1} and U_{2} are disjoint, then we indeed have an "empty union" as referred to in the previous posts (because we don't have any x's).