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.)

]]>Added (here) pointer to

- Rainer M. Vogt,
*Convenient categories of topological spaces for homotopy theory*, Arch. Math 22, 545–555 (1971) (doi:10.1007/BF01222616)

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”.

]]>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)

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

]]>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

]]>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.

]]>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.

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).

]]>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?

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

But please modify as you see the need.

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

]]>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

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

]]>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.

]]>added pointer to:

- Susan Niefield, Section 9 of:
*Cartesianness*, PhD thesis, Rutgers 1978 (proquest:302920643)

added pointer to:

- Francis Borceux, Section 7.2 of:
*Categories and Structures*, Vol. 2 of:*Handbook of Categorical Algebra*, Encyclopedia of Mathematics and its Applications**50**Cambridge University Press (1994) (doi:10.1017/CBO9780511525865)

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?

]]>Thanks for the info!

I made that a footnote (here)

and also added pointer to your article (here).

]]>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. ]]>

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.

]]>