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The Boolean prime ideal theorem and products of cofinite topologies

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Abstract

We show:

  1. The Boolean Prime Ideal theorem math formula is equivalent to each one of the statements:

    1. For every family math formula of compact spaces, for every family math formula of basic closed sets of the product math formula with the fip there is a family of subbasic closed sets math formula (math formula) with the fip such that for every math formula”.
    2. For every compact Loeb space math formula (the family of all non empty closed subsets of math formula has a choice function) and for every set X the product math formula is compact”.
  2. math formula (: 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”.
  3. math formula (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
  4. math formula (: every filter of math formula extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space math formula having a base of size math formula and for every set X of size math formula the product math formula is compact”.

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