If S is a collection of subsets of X and the union of the collection is X, then S is a sub-basis for X.

Note: All you need is this "behavior" to get yourself a basis (by page 82 munkrees). Ask yourself, do these subsets of X combine to make X? Great, then all I need to do is take ALL the finite intersections of these sets and I have a basis (prove this).

IF I want the topology generated by this basis, I can do one of the following:

-Lemma 13.1 : Take the union of all the basis elements.

Also without Lemma 13.1 (or because of Lemma 13.1) the definition of a topology generated by a subbasis is the union of finite intersections of elements of S.