In topology, a compactly generated space or k-space is a topological space whose topology is coherent with the family of all compact subspaces, meaning that for every subset is closed in if and only if is closed in the subspace for all compact subspaces Definitions A subset of a topological space is called (respectively, ) if for every compact subspace is a closed (respectively, open) subset of (when is endowed with the subspace topology induced on it by ). In all topological spaces, every closed subset is always k-closed and every open subset is always k-open, although the converses are not guaranteed in general. Those spaces in which the converses are guaranteed are called compactly generated. Explicitly, a or is a topological space in which every k-closed subset is closed, or equivalently, it is a space in which every k-open subset is open. A is a compactly generated space that is also Hausdorff. This is not to be confused with a Hausdorff-compactly generated space, which may or may not be Hausdorff. Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly Hausdorff. See the category of compactly generated weak Hausdorff spaces for the use in algebraic topology. Coherence with compact subspaces A topological space is said to be with a given family of topological subspaces of (meaning that every is endowed with the subspace topology induced on it by ) if for any subset is closed in if and only if is a closed subset of for every or equivalently, if for any subset is open in if and only if is an open subset of for every This terminology allows the definition of "compactly generated" to be restated as: a space is if it is coherent with the set of all compact subspaces of Thus a topological space is if it satisfies the following condition: for any subset is closed in if and only if is closed in the subspace for all compact subspaces or equivalently, if it satisfies: for any subset is open in if and only if is open in the subspace for all compact subspaces Furthermore, if is coherent with any cover of compact subspaces in the above sense then it is, in fact, coherent with all compact subspaces. This notion of coherence allows the definition of "compactly generated" to be generalized in various ways by changing the family of subspaces.