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 do not.

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.

Let ${\displaystyle (X,T)}$ be a topological space, where ${\displaystyle T}$ is the topology, that is, the collection of all open sets in ${\displaystyle X.}$

There are multiple (non-equivalent) definitions of compactly generated space or k-space in the literature. These definitions share a common structure, starting with a suitably specified family ${\displaystyle {\mathcal {F}}}$ of continuous maps from some compact spaces to ${\displaystyle X.}$ The various definitions differ in their choice of the family ${\displaystyle {\mathcal {F}},}$ as detailed below.

The final topology ${\displaystyle T_{\mathcal {F}}}$ on ${\displaystyle X}$ with respect to the family ${\displaystyle {\mathcal {F}}}$ is called the k-ification of ${\displaystyle T.}$ Since all the functions in ${\displaystyle {\mathcal {F}}}$ were continuous into ${\displaystyle (X,T),}$ the k-ification of ${\displaystyle T}$ is finer than (or equal to) the original topology ${\displaystyle T}$. The open sets in the k-ification are called the k-open sets in ${\displaystyle X;}$ they are the sets ${\displaystyle U\subseteq X}$ such that ${\displaystyle f^{-1}(U)}$ is open in ${\displaystyle K}$ for every ${\displaystyle f:K\to X}$ in ${\displaystyle {\mathcal {F}}.}$ Similarly, the k-closed sets in ${\displaystyle X}$ are the closed sets in its k-ification, with a corresponding characterization. In the space ${\displaystyle X,}$ every open set is k-open and every closed set is k-closed. The space ${\displaystyle X}$ together with the new topology ${\displaystyle T_{\mathcal {F}}}$ is usually denoted ${\displaystyle kX.}$

The space ${\displaystyle X}$ is called compactly generated or a k-space (with respect to the family ${\displaystyle {\mathcal {F}}}$) if its topology is determined by all maps in ${\displaystyle {\mathcal {F}}}$, in the sense that the topology on ${\displaystyle X}$ is equal to its k-ification; equivalently, if every k-open set is open in ${\displaystyle X,}$ or if every k-closed set is closed in ${\displaystyle X;}$ or in short, if ${\displaystyle kX=X.}$

As for the different choices for the family ${\displaystyle {\mathcal {F}}}$, one can take all the inclusions maps from certain subspaces of ${\displaystyle X,}$ for example all compact subspaces, or all compact Hausdorff subspaces. This corresponds to choosing a set ${\displaystyle {\mathcal {C}}}$ of subspaces of ${\displaystyle X.}$ The space ${\displaystyle X}$ is then compactly generated exactly when its topology is coherent with that family of subspaces; namely, a set ${\displaystyle A\subseteq X}$ is open (resp. closed) in ${\displaystyle X}$ exactly when the intersection ${\displaystyle A\cap K}$ is open (resp. closed) in ${\displaystyle K}$ for every ${\displaystyle K\in {\mathcal {C}}.}$ Another choice is to take the family of all continuous maps from arbitrary spaces of a certain type into ${\displaystyle X,}$ for example all such maps from arbitrary compact spaces, or from arbitrary compact Hausdorff spaces.