* Compact subsets of k-spaces (without separation axioms)
@ 2004-02-15 13:47 Gabor Lukacs
2004-02-16 10:30 ` Martin Escardo
0 siblings, 1 reply; 2+ messages in thread
From: Gabor Lukacs @ 2004-02-15 13:47 UTC (permalink / raw)
To: categories
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
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* Re: Compact subsets of k-spaces (without separation axioms)
2004-02-15 13:47 Compact subsets of k-spaces (without separation axioms) Gabor Lukacs
@ 2004-02-16 10:30 ` Martin Escardo
0 siblings, 0 replies; 2+ messages in thread
From: Martin Escardo @ 2004-02-16 10:30 UTC (permalink / raw)
To: Gabor Lukacs, categories
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
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