Let \(X\) be a set. A collection \(\mcb\) of subsets of \(X\) is a basis for a topology on \(X\) if
every \(x\in X\) lies in some \(B\in\mcb\), and
whenever \(x \in B_1 \intersect B_2\) with \(B_1, B_2\in\mcb\), there exists \(B_3\in\mcb\) with \(x\in B_3 \subseteq B_1 \intersect B_2\).
The topology generated by \(\mcb\) is the collection of all \(U\subseteq X\) such that for each \(x\in U\) there exists \(B\in\mcb\) with \(x\in B \subseteq U\).
Let \(\mcb\) be a basis for a topology on a set \(X\). The topology generated by \(\mcb\) is a topology on \(X\), and its open sets are exactly the unions of elements of \(\mcb\).
[@Mun00].