Lemma 13.2: Let X be a topological space. Suppose that C is a collection of open sets of X such that for each open set U of X and each x in U, there is an element c of C such that [itex]x\in c\subset U[/itex]. Then C is a basis for the topology of X. Proof: The first paragraph is trivial, it just shows that the conditions of basis are satisfied. The second paragraph attempts to show that [itex]\tau'[/itex], the topology generated by C, is the same as the topology [itex]\tau[/itex] on X. Can someone elaborate on the second paragraph please!?!