Definition 5.1.11 (Generated Topology).label Let $X$ be a topological space and $\sigma \subset 2^{X}$ be an ideal, then $X$ is $\sigma$-generated if the topology of $X$ is the final topology generated by $\bracs{\iota_S: S \to X|S \in \sigma}$.

If $\kappa \subset 2^{X}$ is the collection of precompact sets of $X$, and $X$ is generated by $\kappa$, then $X$ is compactly generated.