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A topological approach in the extended Fraenkel-Mostowski model of set theory. (English) Zbl 1349.20032

Summary: Lattices of subgroups are presented as algebraic domains. Given an arbitrary group, we define the Scott topology over the subgroup lattice of that group. A basis for this topology is expressed in terms of finitely generated subgroups. Several properties of the continuous functions with respect the Scott topology are obtained; they provide new order properties of groups. Finally there are expressed several properties of the group of permutations of atoms in a permutative model of set theory. We provide new properties of the extended interchange function by presenting some topological properties of its domain. Several order and topological properties of the sets in the Fraenkel-Mostowski model remain also valid in the Extended Fraenkel-Mostowski model, even one axiom in the axiomatic description of the Extended Fraenkel-Mostowski model is weaker than its homologue in the axiomatic description of the Fraenkel-Mostowski model.

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
03E25 Axiom of choice and related propositions
03E30 Axiomatics of classical set theory and its fragments
20B07 General theory for infinite permutation groups
20E25 Local properties of groups
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