function spaces

function spaces

[Martin Escardo]

Dear Comprox and Categories members,

Here is a short note that Reinhold Heckmann and I have written. Your
comments are welcome, as always.

                     On function spaces in topology
                     ------------------------------

    It is the purpose of this expository note to provide a
    self-contained, elementary and brief development of the fact that
    the exponentiable topological spaces are precisely the
    core-compact spaces. The only prerequisite is a basic knowledge of
    topology (continuous functions, product topology and compactness).
    We hope that teachers and students of topology will find this
    useful. As far as we know, there is no such development available
    in the literature.  Although there are one or two embellishments,
    our methods are certainly not original.  We briefly discuss more
    advanced treatments in the introduction.

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http://www.dcs.st-and.ac.uk/~mhe/papers/exponentiablespaces.ps.gz
http://www.dcs.st-and.ac.uk/~mhe/papers/exponentiablespaces.dvi.gz
http://www.dcs.st-and.ac.uk/~mhe/papers/exponentiablespaces.ps
http://www.dcs.st-and.ac.uk/~mhe/papers/exponentiablespaces.dvi
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Best regards,
Martin & Reinhold

Re: function spaces

[Martin Escardo]

I wrote:
>    It is the purpose of this expository note to provide a
>    self-contained, elementary and brief development of the fact that
>    the exponentiable topological spaces are precisely the
>    core-compact spaces. The only prerequisite is a basic knowledge of
>    topology (continuous functions, product topology and compactness).
>    We hope that teachers and students of topology will find this
>    useful. As far as we know, there is no such development available
>    in the literature.  Although there are one or two embellishments,
>    our methods are certainly not original.  
>
>-------------------------------------------------------------------
> http://www.dcs.st-and.ac.uk/~mhe/papers/exponentiablespaces.ps

It turns out, as Fred Linton kindly let me know just after I posted
this, that Eilenberg developed such an account to general function
spaces in topology.  Yesterday I got a copy of Eilenberg's manuscript
(in the literal sense of manuscript) that Fred Linton sent me, which I
read with pleasure. Apparently this will be eventually published. It
was written around 1985. So, after all, there is (going to be) such a
development available in the literature.

The methods that both papers use are the same, and are due to Fox,
Arens, Dugundji, Day and Kelly, Scott, and Isbell (although we combine
them in different ways). These references and most of these methods
are discussed in a paper on function spaces published by Isbell in
1985.

Martin