“In set theory without the axiom of choice but assuming the Boolean prime ideal theorem, every product of nonempty compact Hausdorff spaces is nonempty.”

It turns out that TTcf is equivalent to the Boolean Prime Ideal Theorem (BPI), a priniple well known ( 4 )to be stritly weaker than AC.This note is not atually onerned with Boolean prime ideals. We havementioned BPI only as an identier; it is the most famous of a whole familyof priniples known to be equivalent to one another. Here are four membersof that family: … (TTh) Any produt of ompat Hausdor spaes is ompat.

Theorem 1.1 (Rubin and Scott [8] , Łos and Ryll-Nardzewski [6] ) Products of compact Hausdorff spaces are compact if and only if the Boolean prime ideal theorem holds.

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

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

In other words, working over a choice-free set theory like ZF or even BZ (bounded Zermelo set theory), the ultrafilter principle (UF) implies Tychonoff’s theorem for Hausdorff spaces.

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.

BPI is equivalent to the statement that all products of compact Hausdorff spaces are compact

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: 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 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.

Remarks. (1) That Case 1 in the above proof may occur even if all the Xi’s are non-empty compact Hausdorff spaces is shown by the model N 15 in [12].

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.

Theorem 3 ([8]). The following are equivalent: (i) BPI(ω). (ii) The product 2R is compact.

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,

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.

It is proven that the ultra?lter convergence determines the open sets for every topological space if and only if the*Ultrafilter Theorem*holds.

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.

Indeed, we have BPI => Tych => AC. Thus, the assertion that every product of nonempty compact Hausdorff spaces is nonempty is equivalent to the Boolean prime ideal theorem.

The ultrafilter lemma is equivalent to each of the following statements: … A topological space X {\displaystyle X} is compact if and only if every ultrafilter on X {\displaystyle X} converges to some limit.

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.

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.

The authors also show that the Tihonov product theorem for compact Hausdorff spaces is logically equivalent to some basic theorems in topology and functional analysis, commonly proved using the axiom of choice

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 statement implies the axiom of choice, as proven by Kelley (1950).

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.

The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma.

We use Theorem 2.1(b). Let U be an ultrafilter on X. For each α, the push-forward (πα)∗ (U) is an ultrafilter on Xα. By Theorem 2.1, (πα)∗ (U) & xα for some point xα ∈ Xα. By Theorem 4.2, U & x where x = (xα)α ∈ X. Since every ultrafil ter on X converges, Theorem 2.1 shows that X is compact.

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 $B$.