Compact subsets of k-spaces (without separation axioms)

[Gabor Lukacs <lukacs@mathstat.yorku.ca>]

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