Is there a locally compact space which is not a k-space

Definitions:

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.

• Do you know it must exist, or is it just a hunch? – Henno Brandsma Oct 26 '18 at 17:33
• Well if locally compact implies k space that would be pretty amazing – davik Oct 26 '18 at 17:34
• 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. – Henno Brandsma Oct 26 '18 at 17:38
• The cofinite topology is sequential, thus it is a k-space – davik Oct 26 '18 at 17:41
• Do you know other examples? – 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.