Compactly generated spaces form a convenient category of topological spaces.
The following conditions on a space are equivalent:
A space is a -space if any (hence all) of the above conditions hold. Some authors also say that a -space is compactly generated, while others reserve that term for a -space which is also weak Hausdorff, meaning that the image of any is closed (when is compact Hausdorff). Some authors go on to require a Hausdorff space, but this seems to be unnecessary.
Sometimes -spaces are called Kelley spaces, after John Kelley, who studied them extensively; however, they predate him and the ‘’ does not stand for his name. (Probably it has something to do with ‘compact’ or ‘kompakt’.)
Examples of compactly generated spaces include
Every compact space is compactly generated.
Every locally compact space is compactly generated.
Every topological manifold is compactly generated
Every CW-complex is a compactly generated topological space.
Since a CW-complex is a colimit in Top over attachments of standard n-disks (its cells), by the characterization of colimits in (prop.) a subset of is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the -disks are compact, this implies one direction: if a subset of intersected with all compact subsets is closed, then is closed.
Every first countable space is a compactly generated space.
Since the topology is determined by convergent sequences = maps from one-point compactification ); these include all Fréchet spaces.
Let denote the category of -spaces and continuous maps, and denote the category of all topological spaces and -continuous maps. We have inclusions
The coreflection is denoted , and is sometimes (e.g. by M M Postnikov) also called kaonization and sometimes (e.g. by Peter May) -ification. This functor is constructed as follows: we take as a set, but with the topology whose closed sets are those whose intersection with compact Hausdorff subsets of (the original topology on) is closed (in the original topology on ). Then has all the same closed sets and possibly more, hence all the same open sets and possibly more.
In particular, the identity map 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 is -continuous, so that the counit becomes an isomorphism in . This shows that is essentially surjective, and it is fully faithful since any -continuous function between -spaces is -continuous; hence it is an equivalence.
Since is coreflective, it follows that is complete and cocomplete. Its colimits are constructed as in , but its limits are the -ification of limits in . This is nontrivial already for products: the -space product is the -ification of the usual product topology. The -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.
The categories are cartesian closed. (While in Top only some objects are exponentiable, see exponential law for spaces.) For arbitrary spaces and , define the test-open or compact-open topology on to have the subbase of sets , for a given compact Hausdorff space , a map , and an open set in , where consists of all -continuous functions such that .
(This is slightly different from the usual compact-open topology if 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 itself is Hausdorff, then the two become identical.)
is actually an isomorphism in , which we may call a -homeomorphism (e.g. Strickland 09, prop. 2.12). In fact, it is actually a homeomorphism, i.e. an isomorphism already in .
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 and 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 -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 , , not just -spaces.
Mike Shulman: But any topological space is isomorphic in to its -ification, right? So might as well be defined to consist of -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.
It follows that the category of -spaces and continuous maps is also cartesian closed, since it is equivalent to . Its exponential object is the -ification of the one constructed above for . Since for -spaces, -continuous implies continuous, the underlying set of this exponential space is the set of all continuous maps from to . Thus, when is Hausdorff, we can identify this space with the -ification of the usual compact-open topology on .
Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.
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 by regarding this as the space of maps to the Sierpinski space (the set of truth values in which is closed but not open). From this one can get an exponential law for spaces over if is , so that all fibres of spaces over are closed in their total space. Note that weak Hausdorff implies .
Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that is cartesian closed if is ?
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:
Compactly generated spaces are discussed by J. L. Kelley in his book
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
Other and later references include
George 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.
Samuel Smith, The homotopy theory of function spaces: a survey (arXiv:1009.0804)