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In his 1967 paper A convenient category of topological spaces, Norman Steenrod introduced the category CGH of compactly generated Hausdorff spaces as a good replacement of the category Top topological spaces, in order to do homotopy theory.

The most important difference between CGH and Top is that in CGH there is a functorial homeomorphism $$\mathrm{map}(X,\mathrm{map}(Y,Z))\cong \mathrm{map}(X\times Y,Z),$$ a fact that is only true in Top under the extra assumption that $Y$ is locally compact.

But in more recent papers, I see that people use CGWH spaces instead of CGH spaces... Why?

Could someone explain to me what goes wrong in CGH spaces (please illustrate with an example),
and explain how the "w" fixes everything?

Also (following Jeff's comment), to whom should the "w" be attributed?

One more wish: can someone give me an example of a CGWH space that isn't CGH?

David White
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André Henriques
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    May I add to the question: to whom should the W be attributed? – Jeff Strom Nov 29 '10 at 19:37
  • I'll offer one thought: there's an erratum to one of May's books that I seem to recall consists mainly of "adding the W." I think the issue was that colimits of Hausrorff spaces aren't always Hausdorff. The relevant erratum is on May's webpage, I think. Hopefully I'll find time to give a more directed answer, but it may be a few days. – Dan Ramras Nov 29 '10 at 21:16
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    Well, I think what you should be asking is why use CGWH instead of CG, since after all, compactly generated spaces with no separation axiom are also Cartesian-closed etc. One thing is that compact generation for weakly Hausdorff spaces still takes the "simple form" that the space is the colimit of its compact subsets. For instance, Peter May pointed out to me that the compactly generated Grothendieck topology I introduced on CGH here: http://arxiv.org/abs/0907.3925 extends naturally to CGWH, but, for example, I still don't know how to extend it to CG. – David Carchedi Nov 29 '10 at 21:57
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    This is completely tangential to the question, but I feel obliged to point out some history that I've only become aware of recently: that the fundamental results on cartesian closure of CGH are not due to Steenrod but to Ronnie Brown in his 1961 thesis. The nLab page on convenient categories of topological spaces has recently been updated to include this information; for those interested, I have inserted a link to part A of Brown's thesis in the References. The nLab page is at http://nlab.mathforge.org/nlab/show/convenient+category+of+topological+spaces Comments at the nForum are welcome. – Todd Trimble Nov 29 '10 at 23:20
  • There are some interesting remarks in chapter 1 of May-Sigurdsson on the question of CG (there called "k-spaces") versus CGWH (there called just "CG"). – Mike Shulman Mar 16 '11 at 19:07
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    It's worth mentioning the obvious: philosophically, one should expect CGWH to have nicer properties than CGH because the WH condition (diagonal is closed in the CG topology on the square) is stated in terms of the CG category, whereas the H condition (diagonal is closed in the ordinary product topology) refers back to Top, so there's a "mismatch" in the definition of CGH. It's like defining a scheme to be separated if its underlying space is Hausdorff, which is totally wrong. I would imagine that the pathologies cited in the answers here can be traced back to this mismatch. – Tim Campion Oct 20 '15 at 17:53

6 Answers6

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I believe that CGWH spaces were first used in a systematic way in the work of Lewis-May-Steinberger on spectra. It is certainly the case that Gaunce Lewis's (unpublished) thesis contains the best reference on CGWH spaces that I'm aware of. (I haven't looked at the McCord paper Andrey mentions. Update: Having looked at McCord's paper, it does indeed seem to be the one to introduce CGWH (the idea of which he attributes to J.C. Moore.))

As to why one might prefer to use CGWH spaces, I'm not precisely sure. But here is one possibility.

A key property of the category of CG spaces is that the product of a quotient map with a space is still a quotient map. In CGWH spaces, something even nicer is true: any pullback of a quotient map (along any map) is still a quotient map. (I don't know whether this nicer fact fails in CGH, but I suspect it does.)

Another nice fact about CGWH: regular monomorphisms are precisely the closed inclusions.("Regular monomorphism" means the monomorphism is an equalizer of some pair.) (I originally said here that regular epis in CGWH are precisely quotient maps, but on reflection I'm not sure this is true.)

Charles Rezk
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    That the pullback of a quotient is a quotient sounds strangely close to suggesting that CGWH is locally cartesian closed (and closer still if the pullback of a coequalizer is a coequalizer), and this would be amazing to me. Charles, does Lewis's thesis (or any of the other references) address this issue? – Todd Trimble Nov 29 '10 at 23:30
  • @ Todd, yes that would be amazing. However, I don't think all coequalizers are quotient maps in CGWH (cf Dan Ramras's comment under Harry's answer), so I don't think pullbacks of coequalizers are necessarily coequalizers. – Charles Rezk Nov 30 '10 at 01:22
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    It has always struck me that what these "improved" categories of topological spaces (CG, CGH, CGWH, etc.) are trying to do is to make a version of Top which is as close to having all the properties of a topos as is possible. (And this is why some people prefer to work with simplicial sets instead of spaces; simplicial sets is already a topos!) – Charles Rezk Nov 30 '10 at 01:25
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    Does "quotient" invariably refer to quotient in Top? I would have been inclined to interpret it as a coequalizer of the two projections off an equivalence relation (as interpreted in whatever category), thus a special coequalizer. – Todd Trimble Nov 30 '10 at 01:52
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    Yes, it seems that the chief merit Spanier's quasitopological spaces have is that they are locally cartesian closed. Also worthy of mention is Johnstone's topological topos. – Todd Trimble Nov 30 '10 at 01:55
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    @Charles: I remember some comment made (I think by MJH) about the "convenient category" being either equal to, or related to, a category of sheaves on compact spaces. However, I've never been able to reconstruct the proper statement - is it familiar to you? – Tyler Lawson Nov 30 '10 at 04:02
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    @Tyler: I believe what you're looking for is the following:

    Let $j:CH \hookrightarrow Top$ be the inclusion of compact Hausdorff spaces into topological spaces. It induces a geometric morphism $\left(j_,j^\right)$ between the topoi of sheaves $$Sh\left(CH\right) \to Sh\left(Top\right).$$ The category of compactly generated spaces is equivalent l to the essential image of the restriction of $j^*$ to representable sheaves (i.e.topological spaces). If $Top$ is replaced by Hausdorff spaces or weakly Hausdorff spaces, the analogous statement is also true.

     – David Carchedi May 18 '11 at 16:04
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    @CharlesRezk. "these improved categories of topological spaces (CG, CGH, CGWH, etc.) are trying to do is to make a version of Top which is as close to having all the properties of a topos as is possible". With the advent of condensed sets, we now have a version of Top which is a topos on the nose (with the caveat that there's no set of generators). I guess this means that we could go back and rewrite all the old topology texts with the words 'condensed set' in place of 'CG(W)H space', and all the proofs will work verbatim. – André Henriques Nov 08 '21 at 15:29
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    @AndréHenriques I want to emphasize that condensed sets (i.e. sheaves on compact Hausdorff spaces) already appeared in the comment chain! – user19232801 Mar 23 '23 at 18:23
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A web search suggests that the category of CGWH spaces was introduced in the paper "Classifying Spaces and Infinite Symmetric Products" by M. C. McCord (Transactions of the American Mathematical Society Vol. 146, (1969), pp. 273-298).

McCord motivated introduction of his "weak Hausdorff" separation axiom by noting that

"the requirement of the Hausdorff condition can be a problem because certain standard operations on spaces can lead outside the category", in particular quotient spaces in algebraic topology and topological algebra.

Martin Sleziak
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Andrey Rekalo
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Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$.

It is easy to find such a subspace of the real line. (Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Martin Sleziak
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Paul Fabel
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  • What is the reason $Y$ is CG? It seems any compact subset of $Y$ containing $y$ is finite, and compact subsets not containing $y$ are either finite or sequences converging to $x$ (together with the limit point $x$). Then for fixed $n$ the subset $A_n={y}\cup{1/n+1/(m+n)}$ is k-open (meets every compact in $Y$ is a relatively open set), but not open because its complement is not compact in $X$. What am I missing? – PatrickR May 24 '23 at 05:10
  • $Y$ is compact and hence CG. $X$ is $T_2$ and hence compact subsets of $X$ are closed in $X$. Thus $Y$ is the Alexandroff compactification of $X$. In particular $Y$ is compact and infinite and contains $y$. – Paul Fabel May 24 '23 at 11:18
  • There are different notions of "compactly generated". In particular, CG-1 = Definition 1 and CG-2 = Definition 2 from https://en.wikipedia.org/wiki/Compactly_generated_space. (see also https://topology.pi-base.org/properties?filter=CG). Now CGWH means CG-2 + weak Hausdorff. $Y$ is compact, hence CG-1, but not every compact set is CG-2. That would need to be shown, either directly from the definition, or as a quotient of a locally compact Hausdorff space for example. (see also https://math.stackexchange.com/questions/4646084) – PatrickR May 24 '23 at 22:05
  • You are right that my example does not work. – PatrickR May 26 '23 at 00:42
  • Thanks for pointing out the potential different meanings of CG and the dependence on context. In the example at hand, the starting space is a sequential space (closed sets are precisely those closed under convergent sequences) and convergent sequences have unique limits. – Paul Fabel May 26 '23 at 04:12
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For an example of a CGWH space that is not Hausdorff:

Lamartin (1977) - On the foundations of k-group theory Corollary 1.31 says that if $X$ is CGWH and $A$ is a closed subset of $X$, then the quotient space $X/A$ obtained by collapsing $A$ to a point is also CGWH.

So for any compactly generated space $X$ that is $T_2$ and not $T_3$ (as witnessed by a closed set $A$ which cannot be separated by open sets from some point outside of it), the quotient $X/A$ is CGWH and not Hausdorff. (Any of these spaces will do for $X$.)

PatrickR
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In the [following answer] to a closely related question, Peter May explains a crucial feature of CGWH spaces not shared by CGH spaces.

Community
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André Henriques
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To flesh out my comment above: in the Errata to Geometry of Iterated Loop Spaces (p. 485 here: http://www.math.uchicago.edu/~may/BOOKS/homo_iter.pdf) May states that he should have used weak Hausdorff spaces "in order to validate some limit arguments." I'm not sure exactly what he means; in particular I would think he really means colimit arguments.

Dan Ramras
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