I'm currently looking at Lemma 13.2 in Munkres' Topology. It states the following: Given a collection $C$ of open sets of a topological space $X$ such that for each open set $U$ of $X$ and each $x$ in U, there is an element $C'$ of $C$ such that x $\in C' \subset U$. Then $C$ is a basis for the topology of $X$.
In the proof, it is both shown that $C$ is a basis and that the topology generated by $C$ is equal to the collection of open sets of X. What is the purpose of the second part of the proof? Is it because a particular topology ("the" topology) is specified for X in the lemma? I'm confused because the definition of a basis for a topology on X doesn't mention this.