Contents

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 function $f\colon X \to Y$ between topological spaces is $k$-continuous if for all compact Hausdorff spaces $C$ and continuous functions $t\colon C \to X$ the composite $f \circ t\colon C \to Y$ is continuous.

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

1. For all spaces $Y$ and all functions $f\colon X \to Y$, $f$ 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 subspace $U \subseteq X$ is open if and only if the preimage $t^{-1}(U)$ is open for any compact Hausdorff space $C$ and continuous $t\colon C \to X$.

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 $t\colon C\to X$ is closed (when $C$ is compact Hausdorff). Some authors go on to require a Hausdorff space, but this seems to be unnecessary.

Sometimes $k$-spaces are called Kelley spaces, after John Kelley, who studied them extensively; however, they predate him and the ‘$k$’ does not stand for his name. (Probably it has something to do with ‘compact’ or ‘kompakt’.)

Examples

Examples of compactly generated spaces include

Example

Every compact space is compactly generated.

Example

Every locally compact space is compactly generated.

Example

Every topological manifold is compactly generated

Example

Every CW-complex is a compactly generated topological space.

Proof

Since a CW-complex $X$ is a colimit in Top over attachments of standard n-disks $D^{n_i}$ (its cells), by the characterization of colimits in $Top$ (prop.) a subset of $X$ is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the $n$-disks are compact, this implies one direction: if a subset $A$ of $X$ intersected with all compact subsets is closed, then $A$ is closed.

For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.

Example

Every first countable space is a compactly generated space.

Proof idea

Since the topology is determined by convergent sequences = maps from one-point compactification $\mathbb{N} \cup \{\infty\}$); these include all Fréchet spaces.

Kaonization

Let $k\Top$ denote the category of $k$-spaces and continuous maps, and $\Top_k$ denote the category of all topological spaces and $k$-continuous maps. We have inclusions

$k\Top \to \Top \to \Top_k$

of which the first is the inclusion of a full coreflective subcategory, the second is bijective on objects, and the composite $k\Top \to Top_k$ is an equivalence of categories.

The coreflection $\Top \to k\Top$ is denoted $k$, and is sometimes (e.g. by M M Postnikov) also called kaonization and sometimes (e.g. by Peter May) $k$-ification. This functor is constructed as follows: we take $k(X)=X$ as a set, but with the topology whose closed sets are those whose intersection with compact Hausdorff subsets of (the original topology on) $X$ is closed (in the original topology on $X$). Then $k(X)$ has all the same closed sets and possibly more, hence all the same open sets and possibly more.

In particular, the identity map $id:k(X)\to X$ is continuous, and forms the counit of the coreflection. Thus this coreflection has a counit which is both monic and epic, i.e. a “bimorphism”—such a coreflection is sometimes called a “bicoreflection.”

Moreover, the identity $id: X \to k(X)$ is $k$-continuous, so that the counit becomes an isomorphism in $\Top_k$. This shows that $k\Top \to \Top_k$ is essentially surjective, and it is fully faithful since any $k$-continuous function between $k$-spaces is $k$-continuous; hence it is an equivalence.

Since $k\Top \hookrightarrow \Top$ is coreflective, it follows that $k\Top$ is complete and cocomplete. Its colimits are constructed as in $Top$, but its limits are the $k$-ification of limits in $Top$. This is nontrivial already for products: the $k$-space product $X\times Y$ is the $k$-ification of the usual product topology. The $k$-space product is better behaved in many ways; e.g. it enables geometric realization to preserve products (and all finite limits), and the product of two CW complexes to be another CW complex.

If one is interested in $k$-spaces which are also weak Hausdorff, then there is a further reflector which must be applied; see weakly Hausdorff space.

Properties

Cartesian closure

The categories $k\Top\simeq \Top_k$ are cartesian closed. (While in Top only some objects are exponentiable, see exponential law for spaces.) For arbitrary spaces $X$ and $Y$, define the test-open or compact-open topology on $\Top_k(X,Y)$ to have the subbase of sets $M(t,U)$, for a given compact Hausdorff space $C$, a map $t\colon C \to X$, and an open set $U$ in $Y$, where $M(t,U)$ consists of all $k$-continuous functions $f\colon X \to Y$ such that $f(t(C))\subseteq U$.

(This is slightly different from the usual compact-open topology if $X$ happens to have non-Hausdorff compact subspaces; in that case the usual definition includes such subspaces as tests, while the above definition excludes them. Of course, if $X$ itself is Hausdorff, then the two become identical.)

With this topology, $\Top_k(X,Y)$ becomes an exponential object in $Top_k$. It follows, by Yoneda lemma arguments (prop.), that the bijection

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

is actually an isomorphism in $\Top_k$, which we may call a $k$-homeomorphism (e.g. Strickland 09, prop. 2.12). In fact, it is actually a homeomorphism, i.e. an isomorphism already in $Top$.

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.

Todd Trimble: That may be just the point: that the domain is not necessarily a $k$-space. I have to admit that I haven’t worked through the details of this exposition, but one thing I tripped over is the fact that we’re dealing with all topological spaces $X$, $Y$, not just $k$-spaces.

Mike Shulman: But any topological space is isomorphic in $k\Top$ to its $k$-ification, right? So $k\Top$ might as well be defined to consist of $k$-spaces and continuous maps.

Todd Trimble: Okay, you’re right that makes sense. So in that case, it seems that Zoran definitely has a point here.

Mike Shulman: See the nForum discussion.

It follows that the category $k\Top$ of $k$-spaces and continuous maps is also cartesian closed, since it is equivalent to $\Top_k$. Its exponential object is the $k$-ification of the one constructed above for $\Top_k$. Since for $k$-spaces, $k$-continuous implies continuous, the underlying set of this exponential space $k\Top(X,Y)$ is the set of all continuous maps from $X$ to $Y$. Thus, when $X$ is Hausdorff, we can identify this space with the $k$-ification of the usual compact-open topology on $Top(X,Y)$.

Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.

Local cartesian closure

Unfortunately neither of the above categories is locally cartesian closed.

However, if $K$ is the category of not-necessarily-weak-Hausdorff k-spaces, and $A$ and $B$ are k-spaces that are weak Hausdorff, then the pullback functor $K/B\to K/A$ has a right adjoint. This is what May and Sigurdsson used in their book Parametrized homotopy theory.

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 $\{0,1\}$ 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 $k Top/B$ is cartesian closed if $B$ is $T_0$?

Toby: Hopefully that is explained in the new article.

Mike: Which new article? exponential law for spaces? That page doesn’t talk about fibered exponentials at all.

Toby: Seeing this later, I no longer know what article I meant.

The following article attributes the concept to Hurewicz:

• David Gale, Compact Sets of Functions and Function Rings , Proc. AMS 1 (1950) pp.303-308. (pdf)

Compactly generated spaces are discussed by J. L. Kelley in his book

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

An early textbook account is in

A lectue note careful about the (weakly) Hausdorff assumptions when needed/wanted is in the lecture notes

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

Gabriel and Zisman discuss the connection with the exactness of geometric realization in

• Peter Gabriel, Michel Zisman, Calculus of Fractions and Homotopy Theory , Springer Heidelberg 1967. (ch.III.3-4)

Other and later references include

• Gaunce Lewis, Compactly generated spaces (pdf), appendix A of The Stable Category and Generalized Thom Spectra PhD thesis Chicago, 1978

• George Whitehead, Elements of homotopy theory

• Brian J. Day, Relationship of Spanier’s Quasi-topological Spaces to k-Spaces , M. Sc. thesis University of Sydney 1968. (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, around note 4.3.22 of Algebraic topology from a homotopical viewpoint, Springer (2002) (toc pdf)

• 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

• Samuel Smith, The homotopy theory of function spaces: a survey (arXiv:1009.0804)

• Stefan Schwede, section A.2 of Symmetric spectra (2012)