[FOM] Can We Build Non-trivial Topologies on S4

abuchan@mail.unomaha.edu abuchan at mail.unomaha.edu
Thu Oct 24 16:44:49 EDT 2002

     Consider your favorite axiomitization for S4.  We Know that S4 is
complete with respect to the class of countable reflexive and transitive
trees and subsequently any model of S4 has the so-called tree property (See
Blackburn et al Modal Logic p 353).   Now define a topology t_s4 on <X,R,V>
so that the basic open sets of t_s4 are the upward close subsets of the
S4-tree in question.  This is the rather uninteresting topology know as the
Alexandroff topology. Alexandroff spaces have the property that open sets
are closed under arbritrary intersection.  What I would like to know is it
it is possible to use upward closed sets on the S4-tree as a basis for t_s4
but obtain a space that is at least Hausdorff?  I'm sure that forcing the
pre-oder on the S4-tree into a partial-order will be necessary, and this
will give us T_0.  If anyone can come up with a T_2 space on S4 that uses
some other idea for basic open set, I would be interested in hearing that
as well.

Andrew S Buchan
Department of Mathematics
University of Nebraska
Omaha, NE 68182-0243
TEL: (402) 554-3999
FAX: (402) 554-2975

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