lemma can be used to prove the Hahn-Banach theorem and the Alexander subbase theorem. Applications Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can extend every ideal to a maximal one. This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data. Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with prime filters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements. Both approaches are found in the literature. 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. If we leave out "Hausdorff" we get a theorem equivalent to the full axiom of choice. In graph theory, the de Bruijn–Erdős theorem is another equivalent to BPI. It states that, if a given infinite graph requires at least some finite number in any graph coloring, then it has a finite subgraph that also requires . A not too well known application of the Boolean prime ideal theorem is the existence of a non-measurable set (the example usually given is the Vitali set, which requires the axiom of choice). From this and the fact that the BPI is strictly weaker than the axiom of choice, it follows that the existence of non-measurable sets is strictly weaker than the axiom of choice. In linear algebra, the Boolean prime ideal theorem can be used to prove that any two bases of a given vector space have the same cardinality. See also List of Boolean algebra topics Notes References An easy to read introduction, showing the equivalence of PIT for Boolean algebras and distributive lattices. The theory in this book often requires choice principles. The notes on various chapters discuss the general relation of the theorems to PIT and MIT for various structures (though mostly lattices) and give pointers to further literature. Discusses the status of the ultrafilter lemma. Gives many equivalent statements for the BPI, including