nLab
compactly generated topological space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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.

Definitions

A function f:XYf\colon X \to Y between topological spaces is kk-continuous if for all compact Hausdorff spaces CC and continuous functions t:CXt\colon C \to X the composite ft:CYf \circ t\colon C \to Y is continuous.

The following conditions on a space XX are equivalent:

  1. For all spaces YY and all functions f:XYf\colon X \to Y, ff is continuous if and only if ff is kk-continuous.

  2. There is a set SS of compact Hausdorff spaces such that the previous condition holds for all CSC \in S.

  3. XX is an identification space of a disjoint union of compact Hausdorff spaces.

  4. A subspace UXU \subseteq X is open if and only if the preimage t 1(U)t^{-1}(U) is open for any compact Hausdorff space CC and continuous t:CXt\colon C \to X.

A space XX is a kk-space if any (hence all) of the above conditions hold. Some authors also say that a kk-space is compactly generated, while others reserve that term for a kk-space which is also weak Hausdorff, meaning that the image of any t:CXt\colon C\to X is closed (when CC 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 XX is a colimit in Top over attachments of standard n-disks D n iD^{n_i} (its cells), by the characterization of colimits in TopTop (prop.) a subset of XX is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the nn-disks are compact, this implies one direction: if a subset AA of XX intersected with all compact subsets is closed, then AA 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.

k-Ification

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

kTopTopTop k 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 kTopTop kk\Top \to Top_k is an equivalence of categories.

The coreflection TopkTop\Top \to k\Top is usually denoted kk and sometimes called kk-ification (May 1999, p. 49).

This functor is constructed as follows: we take k(X)=Xk(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) XX is closed (in the original topology on XX). Then k(X)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)Xid: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:Xk(X)id: X \to k(X) is kk-continuous, so that the counit becomes an isomorphism in Top k\Top_k. This shows that kTopTop kk\Top \to \Top_k is essentially surjective, and it is fully faithful since any kk-continuous function between kk-spaces is kk-continuous; hence it is an equivalence.

Since kTopTopk\Top \hookrightarrow \Top is coreflective, it follows that kTopk\Top is complete and cocomplete. Its colimits are constructed as in TopTop, but its limits are the kk-ification of limits in TopTop. This is nontrivial already for products: the kk-space product X×YX\times Y is the kk-ification of the usual product topology. The kk-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 kk-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 kTopTop kk\Top\simeq \Top_k are cartesian closed. (While in Top only some objects are exponentiable, see exponential law for spaces.) For arbitrary spaces XX and YY, define the test-open or compact-open topology on Top k(X,Y)\Top_k(X,Y) to have the subbase of sets M(t,U)M(t,U), for a given compact Hausdorff space CC, a map t:CXt\colon C \to X, and an open set UU in YY, where M(t,U)M(t,U) consists of all kk-continuous functions f:XYf\colon X \to Y such that f(t(C))Uf(t(C))\subseteq U.

(This is slightly different from the usual compact-open topology if XX 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 XX itself is Hausdorff, then the two become identical.)

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

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

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

It follows that the category kTopk\Top of kk-spaces and continuous maps is also cartesian closed, since it is equivalent to Top k\Top_k. Its exponential object is the kk-ification of the one constructed above for Top k\Top_k. Since for kk-spaces, kk-continuous implies continuous, the underlying set of this exponential space kTop(X,Y)k\Top(X,Y) is the set of all continuous maps from XX to YY. Thus, when XX is Hausdorff, we can identify this space with the kk-ification of the usual compact-open topology on Top(X,Y)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 (Cagliari-Matovani-Vitale 95)

However, if KK is the category of not-necessarily-weak-Hausdorff k-spaces, and AA and BB are k-spaces that are weak Hausdorff, then the pullback functor K/BK/AK/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 XX by regarding this as the space of maps to the Sierpinski space (the set {0,1}\{0,1\} of truth values in which {1}\{1\} is closed but not open). From this one can get an exponential law for spaces over BB if BB is T 0T_0, so that all fibres of spaces over BB are closed in their total space. Note that weak Hausdorff implies T 0T_0.

Regularity

The category of compactly generated Hausdorff spaces is a regular category. (Cagliari-Matovani-Vitale 95).

Homotopy

Proposition

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

k(X)wheε X kX,i.e.π 0(k(X))π 0(X),andxXn +(π n(k(X),x)π n(X,x)). 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) \,.

A proof is spelled out here at Introduction to Homotopy Theory.

References

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 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:

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)

Proof that k-spaces form a regular category:

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

See also:


  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.

Last revised on September 7, 2021 at 07:28:10. See the history of this page for a list of all contributions to it.