Compactly generated space

In topology, a topological space ${\displaystyle X}$ is called a compactly generated space or k-space if its topology is determined by compact spaces in a manner made precise below. There is in fact no commonly agreed upon definition for such spaces, as different authors use variations of the definition that are not exactly equivalent to each other. Also some authors include some separation axiom (like Hausdorff space or weak Hausdorff space) in the definition of one or both terms, and others don't.

In the simplest definition, a compactly generated space is a space that is coherent with the family of its compact subspaces, meaning that for every set ${\displaystyle A\subseteq X,}$ ${\displaystyle A}$ is open in ${\displaystyle X}$ if and only if ${\displaystyle A\cap K}$ is open in ${\displaystyle K}$ for every compact subspace ${\displaystyle K\subseteq X.}$ Other definitions use a family of continuous maps from compact spaces to ${\displaystyle X}$ and declare ${\displaystyle X}$ to be compactly generated if its topology coincides with the final topology with respect to this family of maps. And other variations of the definition replace compact spaces with compact Hausdorff spaces.

Compactly generated spaces were developed to remedy some of the shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesian closed category while still containing the typical spaces of interest, which makes them convenient for use in algebraic topology.