9
$\begingroup$

On page 58 of Mark Hovey's book Model Categories, he states the following definitions:

"A subset $U$ of a space $X$ is compactly open if for every continuous $f:K\rightarrow X$ where $K$ is compact Hausdorff, $f^{-1}(U)$ is open in $K$... A space $X$ is called a $k$-space, or Kelley space, if every compactly open subset is open." (emphasis mine)

My question is whether $k$-spaces are called $k$-spaces for John L. Kelley or for some other reason. A quick google search shows me that Kelley studied these spaces a lot, and that he wrote about them as $k$-spaces.'' I interpret this as evidence that they are not named for him, since it's fairly uncommon to hear about a (good) mathematician X going around calling things by his own name. Further evidence for this is a statement an older professor made to me that $k$-spaces were studied by Mac Lane before Kelley.

On the other hand, the word Kelleyfication appears in Mac Lane's Categories for the Working Mathematician (on page 182 of the first edition) as a way to change the topology on a Hausdorff space in order to make it a $k$-space. Furthermore, Mac Lane calls compactly generated Hausdorff spaces Kelley spaces.

1) Can anyone clear this mystery up for me? Does anyone know the first place these spaces appear in the literature, or the first place the category of $k$-spaces was put forth as the ``right'' category of spaces?

2) Is it standard in the literature to assume $k$-spaces are Hausdorff?

Hovey does not, but Mac Lane does. I'm curious about whether there is consensus on this issue.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ Perhaps the $k$ refers to the term "Kaonization" which was apparently used by Postnikov: ncatlab.org/nlab/show/compactly+generated+topological+space $\endgroup$
    – Mark Grant
    Apr 17, 2012 at 7:52
  • 1
    $\begingroup$ The link "wrote about them" has rotted. What did it point to? $\endgroup$
    – LSpice
    Feb 20, 2021 at 17:06
  • $\begingroup$ @LSpice if you hover over it, or click "edit" you can see where it went. You can then try putting that into the wayback machine. I'm unable to remember which specific paper I was linking to 9 years ago. $\endgroup$
    – David White
    Feb 21, 2021 at 13:16
  • $\begingroup$ @DavidWhite, I did try that. The Wayback Machine never crawled it. $\endgroup$
    – LSpice
    Feb 21, 2021 at 15:26

3 Answers 3

Reset to default
12
$\begingroup$

Engelking cites this paper as the place where $k$-spaces were introduced, though the author, David Gale, says the notion was first defined by Hurewicz. The $k$ probably refers to the German `kompakt'.

Improve this answer
$\endgroup$
4
  • 2
    $\begingroup$ Bill Lawvere told me that k definitely doesn't stand for Kelley, and I believe he said that it stands for kompakt. He mentioned work of Gale in the same conversation. I don't remember more details than that. $\endgroup$
    – Tom Leinster
    Apr 17, 2012 at 16:53
  • $\begingroup$ Thanks! I also guessed the $k$ might be for kompakt, but I wasn't certain enough to include that in my question. It's nice to have the reference above and a better sense of the history. I wonder how the notion that $k$ stood for Kelley entered the literature. We may never know. $\endgroup$
    – David White
    Apr 17, 2012 at 23:17
  • $\begingroup$ Name of this paper: Gale - Compact sets of functions and function rings. $\endgroup$
    – LSpice
    Feb 20, 2021 at 17:05
  • $\begingroup$ @DavidWhite Gale's paper was submitted in 1949. According to nforum.ncatlab.org/discussion/8638/… Gale got the notion of k-space from lectures by Hurewicz in 1948-1949. This all predates Kelley's General Topology book of 1955. Kelley's book did popularize the terminology "k-space" (without mentioning its source). Presumably a careless later author inferred from this that "k" stands for Kelley. $\endgroup$
    – PatrickR
    Mar 25, 2023 at 19:37
10
$\begingroup$

Answer to 2). No. Hausdorff means that the diagonal is closed in $X\times X$. The "correct" separation property usually added to $k$-spaces is "weak Hausdorff", which on $k$-spaces means that the diagonal is closed in the $k$-ification of the usual cartesian product.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you for answering. I accepted the other answer since the main part of my question was about where the $k$ came from, but I am certainly glad to have this Hausdorff vs. weak Hausdorff issue cleared up as well. $\endgroup$
    – David White
    Apr 17, 2012 at 23:19
  • 3
    $\begingroup$ What is an example of a space that is weak Hausdorff, but not Hausdorff? $\endgroup$
    – André Henriques
    Apr 28, 2013 at 10:48
  • 1
    $\begingroup$ ( For such an example start with any non locally compact space X such that X is Hausdorff. Then the one point compactification Y is weakly Hausdorff but not Hausdorff.) $\endgroup$
    – Paul Fabel
    Jan 6, 2014 at 15:07
6
$\begingroup$

Just to add a bit to the history, the first time the exponential law was given for Hausdorff k-spaces was I believe in my DPhil thesis, submitted 1961, see PtA available here, which was circulated to the obvious places. In my first paper, (1963), also available here, I wrote:

"It may be that the category of Hausdorff k-spaces is adequate and convenient for all purposes of topology."

In my second paper, (1964), I used the category of Hausdorff spaces and functions continuous on compact subsets, and showed it was what we now called cartesian closed. (My thesis contains an attempt at showing the idea of what we now call monoidal closed, since in my thesis I had lots of internal homs and associated products, usually a tensor.)

I did not understand final topologies at the time and so did not come up with the definition for the non Hausdorff case, but you can find that in my book Topology and Groupoids.

Several people wrote about the non Hausdorff case, but an important application is to fibred exponential laws which were developed by Peter Booth following some ideas sketched by R. Thom. See Booth, Peter I. The section problem and the lifting problem. Math. Z. 121 (1971), 273–287.

I also wonder whether the category defined in

Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.

is indeed adequate and convenient for all purposes of topology, in particular can cope well with fibred exponential laws, since being a topos is a stronger condition.

There are also the purposes of analysis, for which see

Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997.

So the term "convenient" has had a good run!

See also convenient.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.