The Boolean prime ideal theorem and products of cofinite topologies |
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Authors: | Kyriakos Keremedis |
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Affiliation: | University of the Aegean, Department of Mathematics, , Karlovasi Samos, 83200 Greece |
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Abstract: | We show: - The Boolean Prime Ideal theorem
is equivalent to each one of the statements: - “For every family
of compact spaces, for every family of basic closed sets of the product with the fip there is a family of subbasic closed sets ( ) with the fip such that for every ”. - “For every compact Loeb space
(the family of all non empty closed subsets of has a choice function) and for every set X the product is compact”. (: the axiom of choice restricted to families of finite sets) implies “every well ordered product of cofinite topologies is compact” and “every well ordered basic open cover of a product of cofinite topologies has a finite subcover”. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. (: every filter of extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space having a base of size and for every set X of size the product is compact”. |
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Keywords: | Axiom of Choice weak axioms of choice Loeb spaces Tychonoff products Boolean prime ideal theorem E325 54A35 54B10 |
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