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

The construction which follows is basedon example 100 from "Counterexamples inTopology" by Steen and Seebach(ISBN: 0-486-68735-X)Anyone interested in the use of Boolean-valuedforcing models might find this topologyinteresting.  The topology described hereis a semiregular topology.  Steen and Seebachdescribe these topologies as those which aregenerated by a basis consisting of the regularopen sets.The discussion of Boolean-valued models inJech begins with a separative partial orderand performs operations comparable toDedekind cuts in order to obtain a completedBoolean algebra which is described as thealgebra of regular open sets.  Theforcing relation is defined in terms ofthe canonical embedding of a partial orderinto its completion.The first step is to form pairs among theadmissible compositions of connectives.That is, if givenF(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 orderedset<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 productwith the positive integers having the discretetopology.The sense of this is that the connectivity algebrais defined withA(B,C)interpretable as((p B q) A (p C q))In the generation of well-formed formulaeover a countable collection of atomicsentence letters, there will be a countablecollection of formulae corresponding withthe given schema but differing on thebasis of the atomic sentence letters.In addition to the points of this product, thedomain elementsTRUNTRUare adjoined.The topology of interest here is determinedby the product topology arising from theinterval topology and the discrete topologyof the components along with the basisneighborhoodsM_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 topologythat is not compact.
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