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Topology posets and an unramified symmetric model for set theory

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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).