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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.

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Perhaps the $k$ refers to the term "Kaonization" which was apparently used by Postnikov: ncatlab.org/nlab/show/compactly+generated+topological+space –  Mark Grant Apr 17 '12 at 7:52

3 Answers 3

up vote 7 down vote accepted

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'.

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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. –  Tom Leinster Apr 17 '12 at 16:53
    
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. –  David White Apr 17 '12 at 23:17

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.

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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. –  David White Apr 17 '12 at 23:19
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What is an example of a space that is weak Hausdorff, but not Hausdorff? –  André Henriques Apr 28 '13 at 10:48
    
( 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.) –  Paul Fabel Jan 6 at 15:07

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!

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