“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.”

Theorem 3.4 ([2], [3], [10], [12], [14], [22], [25], [28]). Equivalent are: 1. the Tychonoff Theorem for Heine-Borel-compact Hausdorff spaces, … 14. the Boolean Prime Ideal Theorem.

Tychonov’s Theorem (1930): the product of compact topological spaces is compact. This was proved equivalent to AC in Kelley 1950. But for compact Hausdorff spaces it is equivalent to BPI (see below) and hence weaker than AC

The Tychonoff theorem for compact Hausdorff spaces is equivalent to the Boolean algebra prime ideal theorem (see [8] ).

Tychonov's Theorem (1930): the product of compact topological spaces is compact. This was proved equivalent to AC in Kelley 1950. But for compact Hausdorff spaces it is equivalent to BPI (see below) and hence weaker than AC

3. Equivalent are: (a) the Boolean prime ideal theorem, … (e) products of A-compact Hausdorff spaces are A-compact,

Tychonov's Theorem (1930): the product of compact topological spaces is compact. This was proved equivalent to AC in Kelley 1950. But for compact Hausdorff spaces it is equivalent to BPI (see below) and hence weaker than AC

Kelley's specialization of Tychonoff's Theorem is equivalent to the Boolean Prime Ideal Theorem

The principle that “any product of cofinite topologies is compact” is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

The principle that "any product of cofinite topologies is compact" is equivalent (without appealing to the Axiom of Choice) to the Boolean Prime Ideal Theorem.

The Boolean prime ideal theorem or BPIT is equivalent to the ultrafilter principle UF. … The reasoning may be summarized as follows: the ultrafilter principle implies the Tychonoff theorem for compact Hausdorff spaces; see this Remark and the argument immediately preceding it. … This brings us full circle: BPIT implies UF implies Tychonoff(CH) implies BPIT.

The Boolean prime ideal theorem: every proper ideal in a Boolean ring is contained in a prime ideal. … The Tychonoff theorem for Hausdorff spaces: any product of compact Hausdorff spaces is compact; equivalently, any product of compact Hausdorff spatial locales is spatial. (If we drop the Hausdorff condition, then the result is equivalent to the full axiom of choice.)

The assertion that every ideal in a Boolean algebra can be extended to a prime ideal. It is a consequence of the Axiom of choice, but is known to be strictly weaker. It implies the Tikhonov theorem for Hausdorff spaces.

The Boolean prime ideal theorem or BPIT is equivalent to the ultrafilter principle UF. … The reasoning may be summarized as follows: the ultrafilter principle implies the Tychonoff theorem for compact Hausdorff spaces; see this Remark and the argument immediately preceding it. The Tychonoff theorem for compact Hausdorff spaces in turn implies that every Boolean ring BB has a maximal (and therefore prime) ideal; see here. … This brings us full circle: BPIT implies UF implies Tychonoff(CH) implies BPIT.

It can be shown (see John Bell’s online article http://plato.stanford.edu/entries/axiom-choice/here) that BPI is equivalent, under ZF, to some of the well known theorems in mathematics: – Tychonoff’s theorem for Hausdorff spaces: the product of compact Hausdorff spaces is compact under the product topology,

We will show in ZF (i.e., Zermelo–Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI($\omega$). (ii) The Tychonoff product $2^{\mathbb{R}}$, where $2$ is the discrete space $\{0,1\}$, is compact.

However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds.

However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds.

Keywords: Axiom of (Countable) Choice; Boolean Prime Ideal Theorem; Theorems of Ascoli; Baire; Čech-Stone and Tychonoff; compact; Lindelöf and orderable spaces … Summary: Many fundamental mathematical results fail in {\bf{ZF}}, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results --- old and new --- that specify how much ``choice'' is needed {\it precisely} to validate each of certain basic analytical and topological results.

Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent to BPI. For example, the theorem that a product of compact Hausdorff spaces is compact is equivalent to it.

The Boolean prime ideal theorem (BPI) is equivalent, in Zermelo–Fraenkel set theory without the axiom of choice, to Tychonoff's theorem for compact Hausdorff spaces.

Theorem 4.2 which gives prominence to Prime Ideal Theorem defined below is another example of such theorems formulated in terms of the Tychonoff Theorem for Hausdorff spaces.

This is because BPI alone implies that the product of compact Hausdorff spaces is compact (is in fact equivalent to that statement).

An example from topology: $\mathsf{BPI}$ is equivalent to the Tikhonov product theorem for compact Hausdorff spaces, while $\mathsf{AC}$ is equivalent to the full Tikhonov product theorem.

This theorem depends on the Boolean Prime Ideal Theorem (BPI), by way of Topological Space is Compact iff Every Ultrafilter Converges. … Although not as strong as the Axiom of Choice, the BPI is similarly independent of the Zermelo-Fraenkel axioms.

In the other direction we need to use BPI, and we use it in the form of Stone's representation theorem. Given a notion of forcing, we may assume without loss of generality that it is a complete Boolean algebra $B$ and we can consider its Stone space, $S(B)$, the space of all the ultrafilters on $B$.