Compact subsets of k-spaces (without separation axioms)

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