Date: Sat, 12 Aug 1995 23:53:41 -0300 (ADT)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: separate continuity

Date: Fri, 4 Aug 95 15:32:33 EDT
From: Michael Barr <barr@triples.math.mcgill.ca>

Does anyone know anything about topological spaces for which
the topological spaces for which the topology of separate
continuity on their square coincides with the product topology?
No non-discrete T_1 topology is possible; any order topology
has the property and we have at least one other space and that
about exhausts our knowledge of the subject.

Michael

Date: Tue, 15 Aug 1995 14:05:22 -0300 (ADT)
From: categories <cat-dist@mta.ca>
To: categories <categories@mta.ca>
Subject: Re: separate continuity

Date: Mon, 14 Aug 95 19:58:55 +0200
From: Reinhold Heckmann <heckmann@cs.uni-sb.de>

Concerning the question
"Which are the spaces X such that the topology of separate continuity
on X x X coincides with the product topology?"
I know of the following result:
for a T0-space X, the following are equivalent:
1) for every T0-space Y,
   the topology of separate continuity coincides with the product topology
   on X x Y,
2) whenever a point x of X is in an open set U of X,
   there are a finite set F and an open set V of X
   such that   x  in  V  subset  up F  subset  U.
   Here, up F  is
   { b in X | b is above some a in F in the specialization preorder }.

To prove 1 => 2, let Y be the space of open sets of X,
where the topology is generated by assuming all neighborhood filters
of points of X as open.

Every continuous dcpo with its Scott topology satisfies condition (2)
with F being a singleton.
There are non-continuous dcpo's which also do;
called quasi- or multi-continuous by some authors.
With T1, up F = F holds.  Thus, the only T1-spaces satisfying (2)
are the discrete spaces.

The above condition (2) probably is only sufficient,
but not necessary for the special case of X x X.
Hence, this is only a partial answer to the original question.

                                Reinhold Heckmann
                                Universitaet des Saarlandes