Compactly Generated Space

In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:

A subspace A is closed in X if and only if A ∩ K is closed in K for all compact subspaces K ⊆ X.

Equivalently, one can replace closed with open in this definition. If X is coherent with any cover of compact subsets in the above sense then it is, in fact, coherent with all compact subsets.

A compactly generated Hausdorff space is a compactly generated space which is also Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff.