Abstract

We show: The Boolean Prime Ideal theorem BPI is equivalent to each one of the statements: “For every family (Xi=(Xi,Ti))i∈I of compact spaces, for every family G={⋃{πi−1(Fij):i∈Qj}:j∈J} of basic closed sets of the product X=∏i∈IXi with the fip there is a family of subbasic closed sets H (H⊂{πi−1(F):i∈I,Fc∈Ti}) with the fip such that for every j∈J,H∩{πi−1(Fij):q∈Qj}≠⌀”. “For every compact Loeb space Y (the family of all non empty closed subsets of Y has a choice function) and for every set X the product YX is compact”. AC fin (: 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”. CAC fin (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”. BPI(ω) (: every filter of ℘(ω) extends to an ultrafilter) is equivalent to the proposition “for every compact Loeb space Y having a base of size ≤|R| and for every set X of size ≤|R| the product YX is compact”.