Topology posets and an unramified symmetric model for set theory

Thumbnail Image
Date
1988
Authors
Martin, Andrew
Major Professor
Advisor
Alexander Abian
Committee Member
Journal Title
Journal ISSN
Volume Title
Publisher
Altmetrics
Authors
Research Projects
Organizational Units
Organizational Unit
Journal Issue
Is Version Of
Versions
Series
Department
Mathematics
Abstract

A poset (P,≤) with maximum 1 is called compact (C) iff sup W exists and is not 1 for any nonempty well ordered subset W of (P - 1). P is classically compact (CC) iff whenever sup S = 1 for some S ⊆ P, then sup F = 1 for some finite F ⊆ S. The product ∏[superscript] P[subscript] i (ordered coordinatewise) is C (resp., CC) iff the subset of all tuples with finitely many nonunit entries is C (CC). Among other results we show ∏[superscript] P[subscript] i is C(CC) iff each P[subscript] i is C(CC);A poset P is called a T[subscript] i-Topology poset iff P is isomorphic to the open-set lattice of some T[subscript] i-topology. We state necessary and sufficient conditions that a finite poset P must satisfy to be a T[subscript] O, T[subscript] F, T[subscript] Y, T[subscript] DD, T[subscript] FF, or T[subscript]1 topology poset. We show that an infinite poset P is a T[subscript] D-topology poset iff (i) P is a complete distributive lattice (ii) with join-infinite distributivity and (iii) P has a representational witnessed collection [phi] of completely prime filters. [phi] is said to be witnessed iff for any F ϵ [phi] there is an x ϵ F where for any G ϵ [phi] it is the case that x ϵ G iff G is not a proper superset of F;A brief discussion of the set theoretical axioms M and SM is made in order to introduce a method of construction of unramified symmetric models of ZFA + (-AC).

Comments
Description
Keywords
Citation
Source
Subject Categories
Keywords
Copyright
Fri Jan 01 00:00:00 UTC 1988