“The Boolean prime ideal theorem is equivalent, in Zermelo–Fraenkel set theory without the axiom of choice, to Tychonoff's theorem for compact Hausdorff spaces.”
The stated equivalence is a standard result of choice theory. In ZF, the Boolean prime ideal theorem—equivalently, the ultrafilter lemma—proves that products of compact Hausdorff spaces are compact, and that restricted form of Tychonoff’s theorem implies the Boolean prime ideal theorem. Authoritative mathematical sources support both directions.