Compactly generated space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of compactness. View other variations of compactness

Definition

Symbol-free definition

A topological space is said to be compactly generated if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology.

Definition with symbols

A topological space $X$ is said to be compactly generated if there exists a collection $\{K_{i}\}_{i\in I}$ of compact subsets of $X$ , such that a subset $U\subset X$ is open if and only if $U\cap K_{i}$ is open in $K_{i}$ for every $i\in I$ .