nLab
compactly generated topological space

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: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 go on to require a Hausdorff space, but this seems to be unnecessary.

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

Examples

Examples of compactly generated spaces include

Kaonization

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 denoted kk, and is sometimes (e.g. by M M Postnikov) also called kaonization and sometimes (e.g. by Peter May) kk-ification. 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, 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. In fact, it is actually a homeomorphism, i.e. an isomorphism already in TopTop.

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 XX and YY 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 kk-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 XX, YY, not just kk-spaces.

Mike Shulman: But any topological space is isomorphic in kTopk\Top to its kk-ification, right? So kTopk\Top might as well be defined to consist of kk-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 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. 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.

Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that kTop/Bk Top/B is cartesian closed if BB is T 0T_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.

References

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

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

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

  • N. P. Strickland, The category of CGWH spaces, pdf

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

Other and later references include

  • G. Whitehead, Elements of homotopy theory

  • 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)

Revised on April 28, 2013 19:12:13 by Toby Bartels (173.190.128.129)