I came across the following lemma:
Let $X$ be a topological space. Suppose that $\mathcal C$ 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 $\mathcal C$ such that $x \in C \subset U $. Then, $\mathcal C$ is a basis for the topology of $X$
From this Lemma, can we deduce that every topological space $X$ has a basis?
To answer this question, I think all I need to do is to show that the set $\mathcal C$ in the above lemma is non-empty, but it is not clear to me how to prove/disprove this.