nLab
compactly generated space

Idea

A topological space is compactly generated if (in a certain sense) the continuous images in it of all compact Hausdorff spaces tell you everything about its topology.

Compactly generated spaces form a convenient category of topological spaces.

Definitions

A continuous map $f:X\to Y$ of topological spaces is $k$-continuous if for all compact Hausdorff spaces $C$ and continuous functions $t:C\to X$ the composite $f\circ t:C\to Y$ is continuous.

The following conditions on a space $X$ are equivalent:

1. For all spaces $Y$ and all functions $f:X\to Y$, $f$ is continuous if and only if $f$ is $k$-continuous.
2. There is a set $S$ of compact Hausdorff spaces such that the previous condition holds for all $C\in S$.
3. $X$ is an identification space? of a disjoint union of compact Hausdorff spaces.
4. A subset $U\subseteq X$ is open if and only if $t^{-1}(U)$ is open for any compact Hausdorff space $C$ and continuous $t:C\to X$.

A space $X$ is a $k$-space if any (hence all) of the above conditions hold. Some authors also say that a $k$-space is compactly generated, while others reserve that term for a $k$-space which is also weak Hausdorff, meaning that the image of any $t:C\to X$ is closed (when $C$ is compact Hausdorff).

Cartesian closure

The category $kTop$ of topological spaces and $k$-continuous maps is cartesian closed. In fact the exponential map

$$kTop(X\times Y,Z)\to \mathrm{kTop}(X,kTop(Y,Z))$$k\Top(X \times Y, Z) \to kTop(X,k\Top(Y,Z))

is a homeomorphism (not just a $k$-homeomorphism).

Zoran Škoda: I do not understand the remark. I mean if the domain is k-space then by the characterization above continuous is the same as k-continuous. Thus if both domain and codomain are continuous then homeo is the same as k-homeo. I assume that even in noHausdorff case, the test-open topology for $X$ and $Y$ k-spaces gives a k-space and that the cartesian product has the correction for the k-spaces.

The topology on $kTop(X,Y)$ that is used here is the test-open topology, which has the subbase of sets $M(t,U)$ for a given $t:C\to X$ and $U$ open in $Y$ of all $k$-continuous functions $f:X\to Y$ such that $f(t(C))\subseteq U$.

It follows that the category of $k$-spaces and continuous maps is also cartesian closed. This remains true if we also impose the weak Hausdorff condition.

Kaonization

Let us consider for the moment only the categories $\mathrm{Haus}$ of Hausdorff and $\mathrm{kHaus}$ of Hausdorff k-spaces. Then the tautological inclusion $\mathrm{kHaus}\subset \mathrm{Haus}$ has a right adjoint $k$ sometimes (e.g. by M M Postnikov) also called kaonization. This functor is constructed as follows $k(X)=X$ as a set and $k(X)$ has as new closed sets those sets whose intersection with old compacts in $X$ is closed in old topology on $X$. Then $k(X)$ has all the same closed sets and possibly more, hence all the old open sets and possibly more. In particular, the identity map $\mathrm{id}:k(X)\to X$ is continuous. The mapping spaces $\mathrm{kHaus}(X,Y)=k(\mathrm{Top}(X,Y))$ where $\mathrm{Top}(X,Y)$ is the standard mapping space in the sense of compact-open topology. Similarly, the categorical product in $\mathrm{kHaus}$ is the kaonification of the usual (Tyhonov) product. Then $\mathrm{kHaus}$ is cartesianly closed. See G. Whitehead’s Elements of homotopy theory, for more details.

Local cartesian closure

Unfortunately neither of the above categories is locally cartesian closed. There is still a lot of work on fibred exponential laws and their applications. One reason for the success and difficulties is that it is easy to give a topology on the space of closed subsets of a space $X$ by regarding this as the space of maps to the Sierpinski space (the set $\{\mathrm{0,1}\}$ of truth values in which $\{1\}$ is closed but not open). From this one can get an exponential law for spaces over $B$ if $B$ is $T_0$, so that all fibres of spaces over $B$ are closed in their total space. Note that weak Hausdorff implies $T_0$.

Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that $k\mathrm{Top}/B$ is cartesian closed if $B$ is $T_0$?

Toby: Hopefully that is explained in the new article.

References

Compactly generated spaces were first introduced by J. L. Kelley, see his book

• John Kelley?, General topology, D. van Nostrand, New York 1955.

Many properties of compactly generated Hausdorff spaces are used to establish a variant of the theory of fibrations, cofibrations and deformation retracts in

• Norman Steenrod?, A convenient category of topological spaces, Michigan Math. J. 14 (1967) 133–152, project euclid

Later references include

• Ronnie Brown, Topology and groupoids, Booksurge 2006, section 5.9.

• Booth, Peter I.; Heath, Philip R.; Piccinini, Renzo A. Fibre preserving maps and functional spaces. Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158–167, Lecture Notes in Math., 673, Springer, Berlin, 1978.

• Peter May, A concise course in algebraic topology, Chapter 5 file, revised version