The Boolean prime ideal theorem and products of cofinite topologies

We show:

1. The Boolean Prime Ideal theorem is equivalent to each one of the statements:
   1. “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 ”.
   2. “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”.
2. (: 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. (: the axiom of choice restricted to countable families of finite sets) iff “every countable product of cofinite topologies is compact”.
4. (: 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”.