Compact posets and ramifiability of large cardinals
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The main topics of this dissertation are devoted to the study of the partially ordered sets (posets, for short) by using the various consequences of the properties of their order structure in the study of topological spaces and large cardinals;In Section 2, we consider the existence of prime ideals as well as ultrafilters of posets, which play an essential role in the possibility of their embedding into posets in which the order relation is the set-theoretical inclusion. As an application of the results obtained in this Section, we introduce the notion of a subbase S of a poset P and derive some corresponding consequences;In Sections 3 and 4, a key Theorem and a novel technique of transfinite inductive proof are introduced for establishing directly the Tower and Complete Accumulation Point compactness of products of compact topological spaces;The Tower compactness of a topological space T is then proven to be equivalent to the A-inductivity of the poset of all proper open sets of T;In Section 5, coordinatewise construction of complete accumulation points of the countable infinite product of the real unit interval I is given. The results are used in establishing the solvability of infinite systems of linear equations each with at most a countable (finite or infinite) number of unknowns. The existence of a solution of such systems of linear equations over the reals is proved based on the compactness of the product topology and under the assumption that finite subsystem have uniformly bounded solutions;In Section 6, we introduce the concept of Receding sequences of ordinals which serves as the basic motivation for the partition properties of infinite cardinals which in turn lead to the notion of the ramifiablity of infinite cardinals;In Section 7, we consider the question of the existence of ramifiable cardinals and various properties which characterizes them;The existence of a C AC-ramifiable cardinal can be shown in ZFC + MA + ¬CH (where MA is the Martin's axiom). This fact is proved in Section 7 in a rather simple way;In the absence of Martin's axiom, however, a useful characterization of ramifiable cardinals can be derived. To this end in Section 7, we have introduced a special lexicographic order on a Hausdorff cardinal [alpha] which implies that [alpha] is ramifiable. On the other hand, the introduction of a special tree in Section 7 implies the converse provided [alpha] is strongly inaccessible.