Library.

“In standard high-school mathematics and set theory, a set such as A is defined as a collection of individual elements rather than as a single interval object.”

Standard mathematics defines a set as a collection determined by its elements or members. An interval is not an alternative general definition of a set; it is a particular kind of set, usually a subset of the real numbers. A set may still be treated as one unified mathematical object without ceasing to be composed of elements.

“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.