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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) Kelley's `General Topology' 2) nlab 3) Wikipedia.

Definition 1) According to Kelley, X is a k-space means if S is not closed in X, then there exists a closed compact subspace C in X, such that the intersection of C and S is not closed in X. (The Kelley notion of k-space also appears in the paper `Between T-1 and T-2' by Wilansky).

Definition 2) According to n-lab, X is a k-space means if S is not closed in X, there exists a compact Hausdorff space K and a map f:K-->X such that the preimage of S is not compact. This is equivalent to X is CG ( compactly generated) in Neil Strickland's notes on CGWH spaces.

http://ncatlab.org/nlab/show/compactly+generated+topological+space

http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf

Definition 3) Currently 10/03/12, wikipedia has a 3rd notion of k-space and declares that X is a k-space or a CG space provided if S is not closed in X, then there exists a compact subspace C in X, such that the intersection of C and S is not compact.

http://en.wikipedia.org/wiki/Compactly_generated_space

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

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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 3 '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 3 '12 at 22:52
    
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 3 '12 at 23:57
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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. –  Paul Fabel Oct 4 '12 at 3:52
    
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 4 '12 at 3:54

1 Answer 1

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.

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