Contents

# Contents

## Idea

A topological space is called compactly generated – also called a “k-space”1 (Gale 1950, 1., following lectures by Hurewicz in 1948), “Kelley space” (Gabriel & Zisman 1967, III.4), or “kaonic space” (Postnikov 1982, p. 34) – if its topology is detected by the continuous images of compact Hausdorff spaces inside it.

As opposed to general topological spaces, compactly generated spaces form a cartesian closed category while still being general enough for most purposes of general topology, hence form a convenient category of topological spaces (Steenrod 1967) and as such have come to be commonly used in the foundations of algebraic topology and homotopy theory, especially in their modern guise as compactly generated weakly Hausdorff spaces, due to McCord 1969, Sec. 2.

## Definition

###### Definition

($k$-continuous functions)
A function $f\colon X \to Y$ between underlying sets of topological spaces is called $k$-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.

###### Proposition

(equivalent characterizations of compact generation)
The following conditions on a topological space $X$ are equivalent:

1. For all spaces $Y$ and all functions $f \colon X \to Y$ (of underlying sets), $f$ is continuous if and only if $f$ is $k$-continuous (Def. ).

2. There is a set $S$ (instead of a proper class) 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 $t^{-1}(U)$ under any continuous function $t \colon C \to X$ out of a compact Hausdorff space $C$ is open.

###### Definition

(k-spaces)
A topological space $X$ is a $k$-space if any (hence all) of the conditions in Prop. hold.

###### Remark

(terminology)
Some authors say that a $k$-space (Def. ) 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 (especially the early authors on the subject) go on to require a Hausdorff space, but this seems to be unnecessary.

## 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.

## Properties

### Coreflection into topological spaces

###### Definition

(category of k-spaces)
We write

(1)$k\Top \xhookrightarrow{\;\;\;\;} Top$

for the category of $k$-spaces (Def. ) with continuous functions between them, hence for the full subcategory of Top on the k-spaces.

###### Proposition

The inclusion (1) is that of a coreflective subcategory

(2)$k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \bot \;\;\;\;} Top$

The coreflection $k$ is sometimes called $k$-ification (May 1999, p. 49).
###### Proof

The reflection functor $k$ is constructed as follows:

We take $k(X) \coloneqq X$ on underlying sets, and equip this 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 $id \colon k(X)\to X$ is continuous, and forms the counit of the coreflection. Thus this coreflection has a counit which is both a monomorphisms as well as an epimorphisms, i.e. a “bimorphism?”—such a coreflection is sometimes called a “bicoreflection.”

###### Remark

There is also the category $\Top_k$ of all topological spaces and $k$-continuous maps (Def. ). But the composite sequence of inclusions

$k\Top \xhookrightarrow{\;} \Top \xrightarrow{\;} \Top_k \,,$

of which the first is the full inclusion (1) and the second is bijective on objects $k\Top \to Top_k$, is an equivalence of categories.

Namely, the identity morphism $id \colon X \to k(X)$ is $k$-continuous, so that the adjunction counit from Prop. 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.

###### Remark

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 (2) of limits in Top.

Notice that 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.

### Reflection into weak Hausdorff spaces

###### Definition

Write

$h k Top \xhookrightarrow{\;\;\;} k Top$

for the further full subcategory inside that of k-spaces (Def. ) on those which in addition are weak Hausdorff spaces.

###### Proposition

(cgwh spaces reflective in cg spaces)
The full subcategory-inclusion of weak Hausdorff spaces in k-spaces (Def. ) is a reflective subcategory inclusion:

$h k Top \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \bot \;\;\;\; } k Top$

(e.g. Strickland 2009, Prop. 2.22)

###### Remark

(sequence of (co-)reflections)
In summary, Prop. and Prop. yield a sequence of adjoint functors of this form:

$h k Top \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \bot \;\;\;\; } k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \bot \;\;\;\;} Top$

The classical model structure on topological spaces restricts along these (co-)reflective embeddings to a Quillen equivalent model structure on compactly generated topological spaces. See there for more.

### 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

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

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 $Top$.

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.

###### Remark

(failure of local cartesian closure)
Unfortunately neither of the above categories is locally cartesian closed (Cagliari-Matovani-Vitale 95)

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 $T_0$.

### Relation to locally compact Hausdorff spaces

###### Proposition

Every locally compact Hausdorff space is a k-space and is and weakly Hausdorff:

$LCHausSp \xhookrightarrow{\;} h k TopSp$

(Dugundji 1966, XI Thm. 9.3; Strickland 2009, Prop. 1.7)

###### Proposition

The product topological space of a locally compact Hausdorff space with a k-space is already a k-space (i.e. without need of k-ification).

This is proven in Dugundji 1966, XI Thm. 9.4 (also Piccinini 92, Thm. B.4) assuming Hausdorffness, and without that assumption in Escardo, Lawson & Simpson 2004, Cor. 3.4 (iii). Moreover:
###### Proposition

(k-spaces are the colimits in Top of compact Hausdorff spaces)
A topological space is a k-space (Def. ) iff it is a colimit as formed in Top (according to this Prop.) of a diagram of compact Hausdorff spaces.

(Escardo, Lawson & Simpson 2004, Lem. 3.2 (v))

### Relation to compactly generated topological space

Insice $k$-spaces there is the further coreflective subcategory $D Top$ of Delta-generated topological spaces:

$Top \underoverset { \underset{ k }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\bot\;\;\;\;\;\; } k Top_{Qu} \underoverset { \underset{ D }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\bot\;\;\;\;\;\; } D Top_{Qu} \,.$

Both of these coreflections are Quillen equivalences with respect to the classical model structure on topological spaces and the induced model structure on compactly generated topological spaces and the model structure on Delta-generated topological spaces (Gaucher 2009, Haraguchi 2013).

### Regularity

###### Proposition

The categories of

are both regular.

(Cagliari-Matovani-Vitale 95, p. 3).

In particular this implies that in these categories pullback preserves effective epimorphisms (see there).

### Homotopy

###### Proposition

For every topological space $X$, the canonical continuous function from the $k$-ification (the adjunction counit) is a weak homotopy equivalence, hence induces an isomorphism on all homotopy groups:

$k(X) \underoverset {whe} {\;\varepsilon^k_X\;} {\longrightarrow} X \,, \;\;\; \text{i.e.} \;\;\; \pi_0\big(k(X)\big) \xrightarrow{\;\sim\;} \pi_0(X) \,, \;\; \text{and} \;\; \underset{ {x \in X} \atop {n \in \mathbb{N}_+} }{\forall} \Big( \pi_n\big( k(X),\,x \big) \xrightarrow{\sim} \pi_n(X,\, x) \Big) \,.$

E.g. Vogt 1971, Prop. 1.2 (h). The proof is spelled out here at Introduction to Homotopy Theory.

### k-Spaces and CG Hausdorff spaces

The idea of compactly generated Hausdorff spaces first appears in print in:

where it is attributed to Witold Hurewicz, who introduced the concept in a lecture series given in Princeton, 1948-49, which Gale attended.2

Early textbook accounts, assuming the Hausdorff condition:

also:

Influential emphasis of the usefulness of the notion as providing a convenient category of topological spaces:

Early discussion in the context of geometric realization of simplicial topological spaces:

• Saunders MacLane, Section 4 of: The Milgram bar construction as a tensor product of functors, In: F.P. Peterson (eds.) The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday, Lecture Notes in Mathematics 168, Springer 1970 (doi:10.1007/BFb0058523, pdf)

and briefly in

More history and early references, with emphasis on category-theoretic aspects:

• Horst Herrlich, George Strecker, Section 3.4 of: Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 (pdf), pages 255-341 in: C. E. Aull, R Lowen (eds.), Handbook of the History of General Topology. Vol. 1 , Kluwer 1997 (doi:10.1007/978-94-017-0468-7)

The terminology “kaonic spaces”, or rather the Russian version “каонные пространства” is used in

• M M Postnikov, Введение в теорию Морса, Наука 1971 (web)

• M M Postnikov, p. 34 of: Лекции по алгебраической топологии. Основы теории гомотопий, Наука 1982 (web)

Discussion of k-spaces in the generality of subcatgeory-generated spaces, including Delta-generated topological spaces:

Proof that k-spaces form a regular category:

Further accounts:

### CG weak Hausdorff spaces

The idea of generalizing compact generation to weakly Hausdorff spaces appears in:

• Michael C. McCord, Section 2 of: Classifying Spaces and Infinite Symmetric Products, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (jstor:1995173, pdf)

where it is attributed to John C. Moore.

Review in this generality of CG weakly Hausdorff spaces:

Brief review in preparation of the model structure on compactly generated topological spaces:

Review with focus on compactly generated topological G-spaces in equivariant homotopy theory and specifically equivariant bundle-theory:

1. The reason for choosing the term “k-space” in Gale 1950 seems to be lost in history. The “k” is not for “Kelley”, as Kelley 1955 came later. It might have been an allusion to the German word kompakt.

2. This is according to personal communication by David Gale to William Lawvere in 2003, forwarded by Lawvere to Martin Escardo at that time, and then kindly forwarded by Escardo to the nForum in 2021; see there.