this is part of a theorem in my topology book:

A family B of subsets of a set X is a base for a topology of X if and only if B has the following two properties:

1. each x in X lies in at least one set in B

2. if U,V are in B and x is in the intersection of X and V, then there exists W in B such that x is in W and W is a subset of U intersect V.

i was wondering, for #1, shouldnt each x be in T where T is a topology for X? (note: the book states that a base has property 1 because X is open)