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.