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Is there a modern generally accepted answer regarding the notions of k-space or compactly generated space?

For example there are currently at least 3 formally distinct notions of k-space in wide circulation:

  1. In Kelley's General Topology, $X$ is a k-space if for $S \subseteq X$ not closed in $X$ there is a closed compact subspace $C \subseteq X$ such that $C \cap S$ is not closed in $X$.

    (This notion of k-space also appears in A. Wilansky's _Between T1 nd T2 (Amer. Math. Monthly, vol.74, no.3, pp.261-266).)

  2. According to nLab, $X$ is a k-space if whenever $S \subseteq X$ is not closed in $X$, there exists a compact Hausdorff space $K$ and a map $f:K \to X$ such that the preimage of $S$ is not compact.

    This is equivalent to $X$ being compactly generated (CG) in Neil Strickland's note The category of CGWH spaces.

  3. Wikipedia declares that $X$ is a k-space (or a compactly generated space) provided that whenever $S \subseteq X$ is not closed in $X$, then there exists a compact subspace $C$ of $X$, such that the intersection of $C$ and $S$ is not compact.

Are any of definitions 1,2,3 equivalent if $X$ is not weakly Hausdorff?

Martin Sleziak
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Paul Fabel
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  • I don't think it makes sense to define compactly generated to mean anything other than the nLab's definition. – Qiaochu Yuan Oct 03 '12 at 21:50
  • Thanks Qiaochu. I assume by you mean ``is likely to prove most useful''. It is a little unclear if nlab is trying to define k-space or compactly generated or both. The wikipedia definition is suspect. – Paul Fabel Oct 03 '12 at 22:52
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    There’s nothing suspect about the Wikipedia definition: it’s the one found in Willard, for example, and it’s perhaps the most obvious interpretation of the term compactly generated. – Brian M. Scott Oct 03 '12 at 23:57
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    By suspect' I meanhaving good reason for being questioned or challenged' as indicated in the original post. In particular two modern treatises (nlab and Neil Strickland's notes) define compactly generated in a stronger manner than wikipedia. Are they equivalent for general spaces? If not, I would suggest the wikipedia entry is indeed suspect in the sense of being contrary to modern usage. – Paul Fabel Oct 04 '12 at 03:52
  • Moreover the same wikipedia ALSO indicates that compactly generated is equivalent to being a k-space, and wikipedia uses a different definition than the one established in `Kelley'. So that is a 2nd and independent reason for casting suspicion on the wikipedia entry. – Paul Fabel Oct 04 '12 at 03:54
  • I understood what you meant. I disagree. I also don’t much care what nLab says, since category theory mostly leaves me cold. (By the way, it’s only accident that I even saw your comment, since you didn’t include @Brian.) – Brian M. Scott Oct 04 '12 at 20:13
  • The link to N. Strickland's text (The category of CGWH spaces) seems to be dead - but it seems that it can be found elsewhere: https://scholar.google.com/scholar?hl=en&q=Strickland+The+category+CGWH+spaces (At least there is a text with the same filename as the link in the post.) – Martin Sleziak Apr 20 '22 at 14:48

2 Answers2

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I think the Wikipedia definition is not the best, since it does not deal nicely with the non-Hausdorff case, and a quotient of a Hausdorff space need not be Hausdorff. This is kind of related to the question of whether locally compact means each point has a compact neighbourhood, or has a base of compact neighbourhoods. The latter concept is more in tune with the notion of a local property.

A general discussion of "Monoidal closed categories of spaces" is in a paper by Booth and Tillotson (Pacific J Math, vol 88) available here.

Section 5.9 of my book Topology and groupoids (as in the 1988 differently titled edition) has the following result:

5.9.1 Let $X$ be a space. Then the following are equivalent:

(a) $X$ is a $k$-space;

(b) there is a set $\mathcal C_{X}$ of maps $t : C_{t} \to X$ for compact Hausdorff spaces $C_t$ such that a set $A$ is closed in $X$ if and only if $t^{-1}(A)$ is closed in $C_{t}$ for all $t \in \mathcal C_{X}$;

(c) $X$ is an identification space of a space which is a sum of compact Hausdorff spaces.

So the n-lab definition agrees with this. Note that this Section also considers the convenient category of $k$-continuous functions, using the test-open topology on spaces of k-continuous maps.

Actually the idea of fibred exponential laws (i.e. some notion of locally cartesian closed) comes from a paper of Thom on "Homologie des espaces functionels" but the details were sketchy, and were developed by Peter Booth.

Martin Sleziak
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Ronnie Brown
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1

The three definitions are not equivalent for general topological spaces.

As shown in this question, Definition 2 (nlab) implies Definition 3 (Willard), but not conversely. For example the one-point compactification of a space which does not satisfy Definition 2 (like the Arens-Fort space) is compact and hence satisfies Def 3, but does not satisfy Def 2.

Both classes of spaces with Def 2 and with Def 3 are stable under taking quotients, and neither requires the Hausdorff condition. But Def 2 is slightly nicer in that it is stable under taking open sets (see here).

As for Def 1 (Kelley), it seems that it's the least "natural" of the three. Def 1 (resp. Def 3) can be defined as the topology being coherent with the collection of closed compact subspaces (resp. compact subspaces). Being a compact subspace in a space $X$ is an intrinsic property of the subspace. But being compact and closed is not, it depends on how the subspace embeds into $X$. And furthermore, any space is covered by its compact subspaces (which include all singletons). But there need not even exist a covering of the whole space by closed compact subspaces. For example, in the particular point topology with an infinite number of points and particular point $p$, there is no closed compact set containing $p$.

Note that Wilansky's article used Def 1 of k-space, but only in the context of KC spaces (KC = all compacts are closed). For those spaces, Def 1 and Def 3 are equivalent.

PatrickR
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