I am presently reading about the concept of a topological basis, and I seem to be encountering conflicts between various sources. The definition of basis I am working with is the following:
If is a set, a basis for a topology on is a collection of subsets of such that
(1) For each , there exists a basis element such that
(2) If , where and are basis elements, then there is a basis element such that .
Here is where the confusion begins. Some sources (I can't reference any in particular, because I have looked through too many) simply define the topology generated from the basis as that set containing every arbitrary union of elements in , while some (such as Munkres) define it as . Are these definitions equivalent or are they two different notions with a similar name?
I have another question, in regards to the first of the two definitions. If we are only taking unions, how could one obtain the empty set through these unions?