TITLE: Is empty set always part of a basis of a topology?
QUESTION [5 upvotes]: I am reading about point-set topology.
Is empty set always part of a basis of a topology?

REPLY [5 votes]: The important property of a basis is that open sets are exactly those sets that  are unions of basis elements. 
Including the empty set in the basis, or not, does not affect which sets are unions of basis elements, so it doesn't make any difference whether the basis includes the empty set.
(Note that the empty set is always a union of basis elements, since it is the union of no sets.)