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 subspaces in the above sense then it is, in fact, coherent with all compact subspaces.