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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?

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    I don't think it makes sense to define compactly generated to mean anything other than the nLab's definition.2012-10-03
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    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.2012-10-03
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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*.2012-10-03
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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.2012-10-04
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    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.2012-10-04
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    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`.)2012-10-04

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