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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeOct 13th 2010

I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2010

Yes, some clean-up would be nice. I replied to the queries.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeOct 14th 2010

In that case, Zoran’s point stands.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeOct 14th 2010

Looks like the remark in question was originally added by Ronnie Brown over a year ago.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

There is still left-over query box discussion at compactly generated space between Zoran, Todd, Mike and Toby . It looks to me like all issues have been clarified there. Does anyone feel like brushing up the entry?

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

Actually, looking at it again I see maybe an interpretation of the remark that might make it make sense. If by $k Top(X,Y)$ we mean a specific topology which is not necessarily a k-space, then cartesian closure of the category of k-spaces and continuous maps would be about a homeomorphism relating to the k-ifications of this $k Top(X,Y)$ when X and Y are k-spaces, while cartesian closure of the category of all spaces and k-continuous maps could be phrased either as saying the same thing (invoking the fact that they are equivalent) or about a k-homeomorphism relating to $k Top(X,Y)$ itself rather than its k-ification. And in the latter case saying that it is actually a homeomorphism rather than just a k-homeomorphism would be saying something stronger, relating to a characterization of $k Top(X,Y)$ as an arbitrary space rather than (its k-ification) as a k-space. Is this making sense?

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeFeb 23rd 2011

Unfortunately, I am too much in a hectic mode before the long travel next week, so I can not delve into this discussion until about next Thursday…

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeFeb 24th 2011

I’ve edited the entry to clarify some, by distinguishing notationally between the category of k-spaces and continous maps, and the category of all spaces and k-continuous maps. Although they are equivalent, the question at issue seems to revolve around their non-identical-ness.

• CommentRowNumber9.
• CommentAuthorzskoda
• CommentTimeFeb 24th 2011

I still do not have enough time to delve into this but I was told by my student that

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

is extremely clear about when the weakly Hausdorff assumption is needed and when not. I quoted it in $n$Lab.

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeMar 8th 2011

Thanks for the reference; I had a look at it and tried to clarify the entry a trifle further.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeMay 12th 2011
• (edited May 12th 2011)

compactly generated space says that a topological space is compactly generated iff

$X$ is an identification space of a disjoint union of compact Hausdorff spaces.

It is clear to see that a quotient space of a disjoint union of compact Hausdorff spaces if compactly generated, namely every compact Hausdorff space is compactly generated, disjoint union of compactly generated Hausdorff is compactly generated Hausdorff and any quotient of a compactly generated Hausdorff space is compactly generated. Conversely, there are many ideas of covers of $X$ by compact Hausdorff spaces, one needs to choose good enough such that the definition for compactly generated (not necessarily even weakly Hausdorff) space can be tested only on the morphisms from the cover: then one does the identification by identifying points with the same image. However, any attempt which I tried in a quick attempt use the axiom of choice, most often within a proper class or something of the sort. What is the clean set-theoretic way to do it ?

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeMay 13th 2011

What makes you think there is one? (-:

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeMay 14th 2011

So what the entry claims ? It says that it is a characterization of a compactly generated space. Even if we use the axiom of choice is it within a set, or we entail a change of universe or what ? Something is claimed and my attempts to understand it get into set-theoretical nightmare, if I want a full generality.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeMay 17th 2011

Well, I don’t know the answer. I was just saying, just because something is true doesn’t necessarily mean it can be proven in a set-theoretically clean way. (-:

• CommentRowNumber15.
• CommentAuthorzskoda
• CommentTimeMay 17th 2011

I felt that :)

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 13th 2011
• (edited Sep 13th 2011)

I have reorganized the sections at compactly generated space a little (check if you agree that it is better now), added a stubby Examples-section and a reference.

By the way, there is still lots of query-box discussion there. Maybe it can be removed or else turned into definite content?

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeSep 13th 2011

I moved the (still empty) section on “weak Hausdorffification” to weakly Hausdorff space. Maybe k-space and compactly generated space should be separate pages too?

• CommentRowNumber18.
• CommentAuthorKarol Szumiło
• CommentTimeFeb 3rd 2013

In compactly generated topological space there is an inconclusive discussion box about local cartesian closure of the category of compactly generated spaces. I want to bring your attention to this paper I have just stumbled upon

Booth, Peter I. The exponential law of maps. II. Math. Z. 121 (1971), 311–319

where the author claims that the category of compactly generated spaces over a Hausdorff space is cartesian closed. Unfortunately, for the definition of the internal hom-object he refers to another paper of his where the definition is stated in a different language (and involves further references) and I find it difficult to put the definition and the proof together. Perhaps someone will be motivated enough to take a closer look and decide whether the proof is valid.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeApr 25th 2016

I have added in, here, the argument for why every CW-complex is compactly generated.

• CommentRowNumber20.
• CommentAuthorTodd_Trimble
• CommentTimeApr 25th 2016

Returning to #18: I’m fairly skeptical of local cartesian closure. This paper for example says that the category of Kelley spaces (meaning compactly generated Hausdorff spaces) is not locally cartesian closed.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeJun 15th 2016
• (edited Jun 15th 2016)

In the entry where it says “by Yoneda lemma arguments” I have added a pointer to the actual argument (prop.).

Regarding #5 and #6: I vote for simply giving some name such as $[X,Y]$ to the actual exponential object in compactly generated spaces and then simply saying that there is a homeomorphism $[X,[Y,Z]]\simeq [X \times Y, Z]$.

I don’t see how the present state of the entry is trying to improve on this simple fact that the entry should state.

• CommentRowNumber22.
• CommentAuthorTodd_Trimble
• CommentTimeJun 15th 2016
• (edited Jun 15th 2016)

Comment #21 should point here.

• CommentRowNumber23.
• CommentAuthorUrs
• CommentTimeJun 15th 2016

Thanks. Sorry.

• CommentRowNumber24.
• CommentAuthorMike Shulman
• CommentTimeJun 15th 2016

I don’t have the time to re-understand what was going on in #5-6, but I agree that the simple and important facts should be stated up-front in clean notation.

• CommentRowNumber25.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2018

Added a weak form of local cartesian closedness as indicated by Mike Shulman on the nForum.

• CommentRowNumber26.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 18th 2018

There are some old queries on that page that could probably be discarded…

• CommentRowNumber27.
• CommentAuthorUrs
• CommentTimeJun 6th 2020

I just noticed that the section Local Cartesian closure contained the following query box exchange (now moved from there to here) which remains unresolved:

+– {: .query}

Mike Shulman: What precisely does “get an exponential law” mean? Do you mean that $k Top/B$ is cartesian closed if $B$ is $T_0$?

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.

=–

• CommentRowNumber28.
• CommentAuthorTodd_Trimble
• CommentTimeJun 6th 2020

I expect that P. Booth would be an author of such an article. It might be this by Booth and Brown. I can probably firm this up soon.

• CommentRowNumber29.
• CommentAuthorDavidRoberts
• CommentTimeJun 6th 2020
• (edited Jun 6th 2020)

@Todd here’s a better link: https://doi.org/10.1016/0016-660X(78)90049-1, title is Spaces of partial maps, fibred mapping spaces and the compact-open topology, if that rings any bells for anyone.

• CommentRowNumber30.
• CommentAuthorTodd_Trimble
• CommentTimeJun 6th 2020

Thank you David.

• CommentRowNumber31.
• CommentAuthorTim_Porter
• CommentTimeJun 7th 2020

If in doubt ask Ronnie Brown or Peter Booth. At that time (later 1970s early 80s) a lot was done on partial maps.

• CommentRowNumber32.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 5th 2021

Removed an old query box:

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 $X$ and $Y$ 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 $k$-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 $X$, $Y$, not just $k$-spaces.

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

• CommentRowNumber33.
• CommentAuthorRichard Williamson
• CommentTimeApr 5th 2021
• (edited Apr 5th 2021)

[Administrative note: #1 - #24 were originally from another thread with name ’compactly generated space’. Thank you to Dmitri Pavlov for pointing out that the threads should be merged; I have deleted Dmitri’s comment requesting this to avoid confusion in the future, as it now would have been placed out of sync.]

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeApr 15th 2021

both for the claim that the category is not locally Cartesian closed (which was already mentioned, but without reference (nor proof)) and for the claim that it is regular (which I added, thereby cross-linking the comment we had all along at regular category)

• CommentRowNumber35.
• CommentAuthorDmitri Pavlov
• CommentTimeApr 15th 2021

Is there any subcategory of topological spaces that is known to be locally cartesian closed? For example, are Δ-generated topological spaces locally cartesian closed?

• CommentRowNumber36.
• CommentAuthormartinescardo
• CommentTimeApr 16th 2021

Not as far as I know (other than the subcategory of discrete spaces). In this respect supercategories (of subcategories) of topological spaces are much better behaved. A nice example is Johnstone’s topological topos, which embeds all sequential topological spaces. Larger things like that are the new condensed sets and pyknotic sets, which embed compactly generated spaces (which generalize sequential spaces).

• CommentRowNumber37.
• CommentAuthormartinescardo
• CommentTimeApr 16th 2021

The are also the much older quasi-topological spaces which form a quasi-topos modulo your size-view preferences. (And the difference between condensed sets and pyknotic sets is a size preference (in each case).)

• CommentRowNumber38.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021

The entry claims that the category of compactly generated spaces is discussed in

On which page?

I haven’t found it yet, neither scanning through the table of contents nor searching an electronic copy for various choices of related keywords.

• CommentRowNumber39.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021
• (edited Aug 23rd 2021)

Oh, I see: p. 230. Have added this to the reference.

[edit: but that page hardly reflects the “extensive” study that our entry claims Kelley made of k-spaces – where is Kelley’s more extensive writing on the topic? ]

• CommentRowNumber40.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021
• (edited Aug 23rd 2021)

• CommentRowNumber41.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021
• (edited Aug 23rd 2021)

• 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)
• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeAug 24th 2021

(with focus on compactly generated topological G-spaces)

• CommentRowNumber43.
• CommentAuthorUrs
• CommentTimeAug 24th 2021

• 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)
• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

• 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)

as the apparent source of the notion of the “weak Hausdorff”-condition (attributed there to John C. Moore).

• CommentRowNumber45.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

• CommentRowNumber46.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

I have added pointer for the use of “k-ification”.

Our entry also claims the use of “kaonization” and attributes it to Postnikov, but I see almost no usage of “kaonization” and find nothing by Postnikov on the subject. What’s the intended reference?

• CommentRowNumber47.
• CommentAuthorDavid_Corfield
• CommentTimeAug 26th 2021

It’s used in Quantum extensions of ordinary maps, but no information there as to source.

• CommentRowNumber48.
• CommentAuthorUrs
• CommentTimeAug 26th 2021
• (edited Aug 26th 2021)

Yes, that’s the one hit I got which made me write “almost”. But this being from 2018 and not giving a citation, I am inclined to suspect that this author took the terminology from the nLab…

• CommentRowNumber49.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

Also, the term “kaonization” doesn’t really make sense, does it. Even if Postnikov should have used this term somewhere (for which it would still good to have a reference) it might be a good idea not to advertize this usage further.

• CommentRowNumber50.
• CommentAuthorDavid_Corfield
• CommentTimeAug 26th 2021

Presumably it was a joke form of ’k-ification’, like ionization, but its passing won’t be missed. Not that k-ificiation is elegant either.

• CommentRowNumber51.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

Yes. “k-ification” is not elegant, but at least it makes sense as a translated shorthand for “Kompaktifizierung” (which is a standard term, albeit imprecise here), or, rather, something like “Kompaktgenerierung” (which is more precise albeit not standard, but might as well be if history had taken different turns).

• CommentRowNumber52.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 26th 2021
• (edited Aug 26th 2021)

I have definitely seen “kaonization” before. Can try to look up some references later. Concerning etymology, it could be a pun of “canonization”.

• CommentRowNumber53.
• CommentAuthorUrs
• CommentTimeAug 26th 2021

Okay, thanks. No rush, but if you have references, I’d be interested.

• CommentRowNumber54.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 26th 2021

Found it in Postnikov’s book, and I think Postnikov might be the originator of this terminology. http://libgen.rs/book/index.php?md5=34A8C3C956EB80877F4E3CF5A297F514

On page 34 he talks about kaonic spaces (каонные пространства), defined as topological spaces X for which C⊂X is closed if and only if f^* C⊂K is closed for any continuous K→X if K is compact.

On page 33 he talks about kaonic maps (каонные отображения), defined as maps of sets f:X→Y for which K→X→Y is continuous whenever K→X is continuous and K is compact.

On page 387 he uses this terminology again.

Postnikov also uses the same terminology more extensively (over a dozen occurrences) in the book http://libgen.rs/book/index.php?md5=4BF450585846A0531FF485E34D062C0A.

Another source I was able to find is Postnikov’s translation of the book by Gabriel and Zisman into Russian: http://libgen.rs/book/index.php?md5=84F649B0DEE2C53F4101735ABE4ED8BE

• CommentRowNumber55.
• CommentAuthorUrs
• CommentTimeAug 27th 2021

Thanks!

I have added these pointers to the entry (here).

Do you see if, apart from “kaonic spaces” and “kaonic maps” he speaks of “kaonization”?

Do you know if any of this made it into English publications?

By the way, while we are hunting references: We are also still looking for a reference that would reflect Kelley’s “extensive” study of CG-spaces, as claimed by the entry. The reference Kelley 1955 seems to have only a single page on them, with just the basic definitions.

• CommentRowNumber56.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 27th 2021

Re #55: In a sense it did, given that Andre Kornell does use “kaonization” extensively in his published papers. I could not find specific occurrences of “каонизация” in Postnikov’s writings, though.

Concerning “Kelley spaces”: this terminology appears to originate in Calculus of Fractions and Homotopy Theory by Gabriel and Zisman, where it is used all over the place.

They do not give any references, but I presume they refer to his book General Topology, which is one of the earliest sources for them.

Kelley does perform nontrivial things with k-spaces, see, for example, his version of the Arzela-Ascoli theorem for k-spaces on page 234 of his book.

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeAug 27th 2021
• (edited Aug 27th 2021)

Thanks again!

This made me break up the discussion of terminology into two items

• terminology for the spaces themselves – now here.

• terminology for the reflection operation – now here.

Postnikov is now cited for the former, no longer for the latter.

Kornell does use “kaonization” extensively in his published papers.

For better or worse. But how extensively really? Google gives me a single hit: arXiv:1811.01922, and that is – let’s beware – from over 9 years after the nLab started to – apparently incorrectly – proclaim this terminology in rev 8 (from Zoran).

It seems likely to me that an author who starts looking into the subject only from after 2009 will have picked up that terminology from the $n$Lab, where it was (until I finally changed it the other day) no less but a section title.

I fear we may just perpetuate a mistake that we were responsible for in the first place if we give any weight to “kaonization”.

But please let me know how extensively you feel Kornell has really been using it, meanwhile.

• CommentRowNumber58.
• CommentAuthormartinescardo
• CommentTimeAug 27th 2021
• (edited Aug 27th 2021)
One more reference:

D. Gale, Compact sets of functions and function rings, Proc. Amer. Math. Soc. 1 (1950),
303–308.

There Gale proves the Arzela-Ascoli theorem for k-spaces for the first time.

According to Gale, he learned about k-spaces from Hurewicz in lectures in Princeton in 1948-1949. I learned this from Lawvere in 2003. He said to me "Now I have telephoned David Gale again. He states that he had participated in a seminar at Princeton in 1948-1949 in which Hurewicz lectured on his k-space definition and on the exponential law which results.(He seemed gratified to hear that people are still benefiting from his writings of 54 years ago). Kelley and Gale certainly knew each other personally since they were both at Berkeley for many years."

This was when I was trying to clarify the history of compactly generated spaces for a paper I wrote about them with Jimmie Lawson and Alex Simpson.
• CommentRowNumber59.
• CommentAuthorUrs
• CommentTimeAug 27th 2021

Thanks for the info!

I made that a footnote (here)

• CommentRowNumber60.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 27th 2021

Re #57: “Kaonization” in Kornell’s paper may well be an example of a citogenesis, just like “étalé space” in Carchedi’s paper. We could ask him.

Concerning Zoran’s contribution to Revision 8, can we ask him directly?

• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeAug 28th 2021

• CommentRowNumber62.
• CommentAuthorUrs
• CommentTimeAug 28th 2021

• CommentRowNumber63.
• CommentAuthormartinescardo
• CommentTimeSep 2nd 2021
Regarding #59, Urs, I think it is a very good idea to write down that Hurewicz is the originator of the notion and theory of compactly generated spaces, and attribute this to Gale and Lawvere. However, I feel that copying literally Lawvere's message in the page may be too personal for an nlab page consulted by thousands of students and mathematicians?
• CommentRowNumber64.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021

Hi Martin,

I have taken this verbatim from the comment you made public in #58. Do you want to keep that comment #58 public? If not, you can edit/clear it any time by clicking “edit” in the top right.

Do we have another source to use as the citation for the claim that Hurewicz gave the definition in lectures in Princeton 1948?

Does anyone know how to contact David Gale? (Searching for him seems almost impossible due to prominent namesakes.) If I had an email address, I would contact him to get this info first hand.

• CommentRowNumber65.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021

I have added (here) the statement that $k$-ifications $X \to k(X)$ are weak homotopy equivalences.

1. The subcategory of k-spaces is coreflective (the k-ification has more opens), and the counit is a weak homotopy equivalence. (The unit is, too, of course :))

Anonymous

• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeSep 2nd 2021

Woops. Thanks for catching. As you noticed, I had this mixed up already in the lecture notes statement pointed. Thanks for fixing!

• CommentRowNumber68.
• CommentAuthormartinescardo
• CommentTimeSep 2nd 2021
Hi Urs, unfortunately Gale passed away in 2008. I think it would be fine to attribute the conversation to Lawvere in the page, without mentioning the email message, and keep the message here. I will edit the page.
• CommentRowNumber69.
• CommentAuthormartinescardo
• CommentTimeSep 2nd 2021
Oh, somebody seems to be editing the page. I will edit it later to replace footnote 2 to say "according to personal communication to Lawvere from David Gale in 2003", if you agree. Without the reference to this discussion page. But I would keep the email message quotation here as a kind of a historical record.
• CommentRowNumber70.
• CommentAuthorUrs
• CommentTimeSep 3rd 2021

Okay, I have further adjusted the footnote (see here).

But please modify as you see the need.

• CommentRowNumber71.
• CommentAuthormartinescardo
• CommentTimeSep 7th 2021
Looks good to me. Thanks, Urs.
• CommentRowNumber72.
• CommentAuthorUrs
• CommentTimeSep 7th 2021

Okay, thanks. It looks like, with your help, we saved this information last-minute from disappearing into forgottenness.

By the way, I used to assume that this David Gale is not the David Gale who wrote about k-spaces – but since you said above that our David Gale passed away in 2008, maybe that’s the same person after all? Do you know?

• CommentRowNumber73.
• CommentAuthorDavid_Corfield
• CommentTimeSep 7th 2021
• (edited Sep 7th 2021)

Seems very likely. The Wikipedia Gale worked on games, but was supervised by the topologist Tucker, who incidently also supervised Isbell (and John Nash and Marvin Minsky).

• CommentRowNumber74.
• CommentAuthorUrs
• CommentTimeSep 7th 2021
• (edited Sep 7th 2021)

Okay, I found the proof: This “MacTutor biography” clarifies it. It’s the only source that I have seen so far which admits that this David Gale authored “Compact sets of functions and function rings”.

Okay, great, have created a category:people-entry David Gale now.

• CommentRowNumber75.
• CommentAuthorUrs
• CommentTimeSep 21st 2021
• (edited Sep 21st 2021)

I have tried to give more logical structure to the (old) section on the coreflection (now here) by adding numbered environments and cross-pointers, and disentangling the construction of the coreflection (which is still lacking a word on why it’s actually a coreflection) from the discussion of the equivalence $k Top \simeq Top_k$.

In the course of doing this I ended up doing similar edits to the Defintion section here (adding numbered environments and cross-pointers).

Finally, I have started a new Properties-section “Reflection into weak Hausdorff spaces” (here), showing the other adjunction.

In doing so, I have stuck – for the time being – with the notation “$k Top$” long used in the previous section, and so now $Top_{CGWH}$ is denoted “$h k Top$” here. I am not saying this is a good idea, necessarily, but it’s momentarily the least disruptive to the old material in the entry.

• CommentRowNumber76.
• CommentAuthorUrs
• CommentTimeSep 22nd 2021
• (edited Sep 22nd 2021)

I have started a new Properties-subsection “Relation to locally compact Hausdorff spaces” (here)

Regarding the statement that a Hausdorff space is a k-space iff it is the quotient of a locally compact Hausdorff space:

Is that quotient then also the coequalizer in weak Hausdorff spaces (?) Need to think about this tomorrow when I am more awake…

$\,$

also, I have split the References-section into two subsections, and re-organized slightly, for more systematics

• CommentRowNumber77.
• CommentAuthorUrs
• CommentTimeSep 23rd 2021
• (edited Sep 23rd 2021)

I have added boosted versions of the statements about quotients/colimits of LCH spaces here and here, following discussion in another thread (here)

• CommentRowNumber78.
• CommentAuthorUrs
• CommentTimeSep 29th 2021

added this pointer, for relation to $\Delta$-spaces:

• Philippe Gaucher, Section 2 of: Homotopical interpretation of globular complex by multipointed d-space, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (arXiv:0710.3553)
• CommentRowNumber79.
• CommentAuthorUrs
• CommentTimeSep 30th 2021

and grouped this together with Escadro-Lawson & Simpson 2004 and with Gaucher 2009 under “discussion in the generality of subcategory-generated spaces, including $\Delta$-generated spaces”.

• CommentRowNumber80.
• CommentAuthorUrs
• CommentTimeNov 9th 2021
• (edited Nov 9th 2021)

I have expanded the statement of regularity (here) by saying that not only compactly generated Hausdorff spaces but also cg weakly Hausdorff spaces form a regular category.

(This is from the same page 3 in Cagliari, Matovani and Vitale 1995, in the Remark below the Theorem.)