Date: Dec 7, 2012 5:31 AM
Author: fom
Subject: fom - 08 - topologizing the connectivity algebra


The construction which follows is based
on example 100 from "Counterexamples in
Topology" by Steen and Seebach
(ISBN: 0-486-68735-X)

Anyone interested in the use of Boolean-valued
forcing models might find this topology
interesting. The topology described here
is a semiregular topology. Steen and Seebach
describe these topologies as those which are
generated by a basis consisting of the regular
open sets.

The discussion of Boolean-valued models in
Jech begins with a separative partial order
and performs operations comparable to
Dedekind cuts in order to obtain a completed
Boolean algebra which is described as the
algebra of regular open sets. The
forcing relation is defined in terms of
the canonical embedding of a partial order
into its completion.



The first step is to form pairs among the
admissible compositions of connectives.
That is, if given


F(x,y)


it is to be paired with


(F*F)(x,y)


where * is the multiplicative product.



The next step is to form the linearly ordered
set


<F(x,y), G(u,v), H(m,n), ..., *, ..., (H*H)(m,n), (G*G)(u,v), (F*F)(x,y)>


indexed numerically according to


<1, 2, 3, ..., omega, ..., -3, -2, -1>


and give it the interval topology. Then form a product
with the positive integers having the discrete
topology.

The sense of this is that the connectivity algebra
is defined with


A(B,C)

interpretable as


((p B q) A (p C q))


In the generation of well-formed formulae
over a countable collection of atomic
sentence letters, there will be a countable
collection of formulae corresponding with
the given schema but differing on the
basis of the atomic sentence letters.



In addition to the points of this product, the
domain elements

TRU

NTRU

are adjoined.


The topology of interest here is determined
by the product topology arising from the
interval topology and the discrete topology
of the components along with the basis
neighborhoods

M_n+(TRU) = ({TRU} u {(x_i,j) | i<omega, j>n})

M_n-(NTRU) = ({NTRU} u {(x_i,j) | i>omega, j>n})



This is a semiregular Hausdorff topology
that is not compact.