Editing Boolean prime ideal theorem (section)

==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 [[Tychonoff's theorem|theorem]] equivalent to the full axiom of choice.

In [[graph theory]], the [[De Bruijn–Erdős theorem (graph theory)|de Bruijn–Erdős theorem]] is another equivalent to BPI. It states that, if a given infinite graph requires at least some finite number {{mvar|k}} in any [[graph coloring]], then it has a finite subgraph that also requires {{mvar|k|colors}}.<ref>{{citation
 | last = Läuchli | first = H.
 | year = 1971
 | title = Coloring infinite graphs and the Boolean prime ideal theorem
 | journal = [[Israel Journal of Mathematics]]
 | doi = 10.1007/BF02771458 | doi-access=
 | mr = 0288051
 | s2cid = 122090105
 | volume = 9
 | issue = 4
 | pages = 422–429
 }}</ref>

A not too well known application of the Boolean prime ideal theorem is the existence of a [[non-measurable set]]<ref>{{citation
 | last = Sierpiński | first = Wacław | author-link = Wacław Sierpiński
 | year = 1938
 | title = Fonctions additives non complètement additives et fonctions non mesurables
 | language = French
 | journal = [[Fundamenta Mathematicae]]
 | doi = 10.4064/fm-30-1-96-99 | doi-access = free
 | volume = 30
 | pages = 96–99
 }}</ref> (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 [[Basis (linear algebra)|bases]] of a given [[vector space]] have the same [[cardinality]].