A locally compact space is a space where every point has a local base of compact neighborhoods.

A $k$-space $X$ has its topology generated by maps from compact Hausdorff spaces, i.e. $C$ is closed iff for every compact Hausdorff space $K$ and every continuous function $f: K \to X$, $f^{-1}[C]$ is closed in $K$. Strickland's notes call this compactly generated.

By compact I mean not necessarily Hausdorff.

The reason I ask is that standard constructions of non $k$-spaces for example the square of the one-point compactification of $\mathbb{Q}$ and the product $\mathbb{R}\setminus \{1,\frac{1}{2},\frac{1}{3}\} \times \mathbb{R}/\mathbb{Z}$ where the second quotient means identifying $\mathbb{Z}$ to one point, are usually not locally compact.

I'm looking for a locally compact space which is not a $k$-space.

  • $\begingroup$ Do you know it must exist, or is it just a hunch? $\endgroup$ – Henno Brandsma Oct 26 '18 at 17:33
  • $\begingroup$ Well if locally compact implies k space that would be pretty amazing $\endgroup$ – davik Oct 26 '18 at 17:34
  • $\begingroup$ How many locally compact non-Hausdorff spaces do you know off? Cofinite is one that comes to mind but I think that it’s not a $k$-space in your sense. $\endgroup$ – Henno Brandsma Oct 26 '18 at 17:38
  • $\begingroup$ The cofinite topology is sequential, thus it is a k-space $\endgroup$ – davik Oct 26 '18 at 17:41
  • $\begingroup$ Do you know other examples? $\endgroup$ – Henno Brandsma Oct 26 '18 at 17:43

https://en.wikipedia.org/wiki/Compactly_generated_space In the example section it is said that locally compact space are $k$-space.

  • $\begingroup$ Yeah this is one of the problems with terminology which is why I took so long defining all the terms lol. Wikipedia defines k space differently so it's sort of obvious in their definition that it includes locally compact spaces. Whereas for me LCH is k space but without Hausdorff it's not obvious $\endgroup$ – davik Oct 25 '18 at 22:47

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