From that information we cannot determine which of the two alternatives A or B is true. The Boolean Prime Ideal Theorem and the Axiom of Dependent Choice, discussed briefly below, are known to be strictly weaker than the Axiom of Choice.
Any net has a universal subnet.
Any completely regular Hausdorff space has a Stone-Cech compactification. Any product of compact Hausdorff spaces is compact. Surely the most important of these is the Baire Category Theorem. However, this is only a first approximation, since it still permits the Axiom of Foundation and proofs by contradiction. The latter notion has the advantage that it is very precise and can be studied with some success using fairly simple mathematical tools, discussed below.