What is basis of a topology?

A basis of $\mathcal T$ is a family $\mathcal B \subseteq \mathcal T$ such that every open set is a union of members of $\mathcal B$ (e.g. open balls in a metric space; open intervals in $\R$). A family $\mathcal B$ of subsets of $X$ is a basis of some topology iff it covers $X$ and, for $B_1, B_2 \in \mathcal B$ and $x \in B_1 \cap B_2$, some $B_3 \in \mathcal B$ has $x \in B_3 \subseteq B_1\cap B_2$ — then “unions of members” is a topology, the topology generated by $\mathcal B$.