Definition 4.1.6 (Generated Topology). Let $X$ be a set and $\ce \subset 2^{X}$ such that $\bigcup_{U \in \ce}U = X$, then the smallest topology on $X$ containing $\ce$ is given by
\[\topo(\ce) = \bracs{\bigcup_{i \in I}U_i \bigg | U_i \in \cb(\ce)}\]
where
\[\cb(\ce) = \bracs{\bigcap_{j = 1}^n U_j \bigg | \seqf{U_j} \subset \ce, n \in \nat^+}\]
is a base for $\topo(\ce)$. The topology $\topo(\ce)$ is known as the topology generated by $\ce$.
Proof. Since $\cb(\ce) \supset \ce$, $\cb(\ce)$ satisfies (TB1). In addition, $\cb(\ce)$ is closed under finite intersections, so it satisfies (TB2). By Definition 4.1.5, $\cb(\ce)$ is a base for $\topo(\ce)$.$\square$