Compact subsets of k-spaces (without separation axioms)

Compact subsets of k-spaces (without separation axioms)

[Gabor Lukacs]

Dear Topologists, Categorists and Categorical Topologists,


It is well-known that for a *Hausdorff* topological space X, its
k-ification kX and X have the same compact subspaces.

It is also well-known that when we assume no separation axioms, X and kX
have the same k-continuous maps (i.e. maps f:X -->Y such that ft is
continuous for every "test-function" t: K --> X, where K is a compact
Hausdorff space).

I was wondering if anyone knows whether the first statement is true
*without separation axioms*, i.e., whether for every topological space X,
its k-ification kX and X have the same compact subspaces.

I would very much appreciate any suggestion, reference or counterexample.


Gabor Lukacs

Re: Compact subsets of k-spaces (without separation axioms)

[Martin Escardo]

Gabor Lukacs writes:
 > I was wondering if anyone knows whether the first statement is true
 > *without separation axioms*, i.e., whether for every topological space X,
 > its k-ification kX and X have the same compact subspaces.

Sometime ago Alex Simpson advertised a paper

"Comparing cartesian closed categories of (core) compactly generated
spaces". http://www.cs.bham.ac.uk/~mhe/papers/ELS03.pdf

where we have the same question (problem 9.2). Hence if anyone has an
answer, please forward it to us as well - thanks.

(I also take the opportunity to mention that we have updated the paper
with some references given to us by some members of this list, in
Section 3.)

MHE