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Forcing in set theory and its applications to topology

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This dissertation consists of two parts. The first seven sections are concerned with Set Theory and the last four with Topology. These parts are linked by the recent developments in Axiomatic Set Theory. In the first part of our work we develop a new forcing language and use the notion of the hereditarily symmetric sets to prove the relative consistency of ZF('-) + (IL-PERP)AC. Our method consists in the reduction of a model of ZF('-) to that of ZF('-) + (IL-PERP)AC;In the first part, we also develop a variant of the unramified forcing language defined by the following five clauses. (UNFORMATTED TABLE FOLLOWS); (1) p a (ELEM) b iff ((FOR ALL)h)(h (LESSTHEQ) p (--->) ((THERE EXISTS)c)((THERE EXISTS)m (LESSTHEQ) h); ((THERE EXISTS)q (GREATERTHEQ) m)((c,q) (ELEM) b (WEDGE) m a = c); (2) p a = b iff ((FOR ALL)h)(h (LESSTHEQ) p (--->) (IL-PERP)((THERE EXISTS)c)(h; ((c (ELEM) a (WEDGE) c (NOT ELEM) b) V (c (NOT ELEM) a (WEDGE) c (ELEM) b))); (3) p (IL-PERP)Q iff ((FOR ALL)h)(h (LESSTHEQ) p (--->) (IL-PERP)(h Q)); (4) p (Q V S) iff (p Q) V (p S);(5) p ((THERE EXISTS)x)Q(x) iff ((FOR ALL)h)(h (LESSTHEQ) p (--->) ((THERE EXISTS)m)(m (LESSTHEQ) h (WEDGE); ((THERE EXISTS)b)(m Q(b))).(TABLE ENDS);We show that the forcing as defined in (1)-(5) preserves the law of the double negation;In Section 9 of the second part, we introduce the E-spaces which satisfy separation conditions weaker than normality. Then using Martin's and generalized Martin's axioms we prove some theorems concerning normality of E-spaces. In particular in Theorem 31 we show under GMA + CH that if a space X has a dense subset of cardinality (omega)(,1), every infinite subset of which has an A-point, then every open cover of cardinality less than 2('(omega))1, contains a countable subcollection whose union is dense in X. Later we prove under MA + (IL-PERP)CH that a regular, E(,3), hereditarily separable, countably compact space X, having a base of <2('(omega)) many open sets, is a T(,4) space. We also prove under GMA + CH, a result similar to the last one. In Section 10 based on cardinality considerations we introduce some weaker versions of compactness and prove the various implications that hold among them. We also introduce some new topologies on the Cartesian product of a family of spaces and then show that some of the notions of compactness are preserved under products of certain cardinalities.