Contents

# 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 $T_0$.

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)