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?

  • $\begingroup$ I don't think it makes sense to define compactly generated to mean anything other than the nLab's definition. $\endgroup$ – Qiaochu Yuan Oct 3 '12 at 21:50
  • $\begingroup$ 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. $\endgroup$ – Paul Fabel Oct 3 '12 at 22:52
  • $\begingroup$ 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. $\endgroup$ – Brian M. Scott Oct 3 '12 at 23:57
  • 3
    $\begingroup$ By suspect' I mean having 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. $\endgroup$ – Paul Fabel Oct 4 '12 at 3:52
  • $\begingroup$ 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. $\endgroup$ – Paul Fabel Oct 4 '12 at 3:54

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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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