Page "Axiom of choice" ¶ 96
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## Some Related Sentences

** and theorem
** Well-ordering theorem: Every set can be well-ordered.
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.
** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
** The Nielsen – Schreier theorem, that every subgroup of a free group is free.
** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals
** The Banach – Alaoglu theorem about compactness of sets of functionals.
** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.
** Hilbert's basis theorem
** Bayes ' theorem
** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.
** Lyapunov's central limit theorem
** Superposition theorem, in electronics
** " Kelvin's vorticity theorem for incompressible or barotropic flow ".
** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem
** Various proofs of the four colour theorem.

** and every
** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.
** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
** For every non-empty set S there is a binary operation defined on S that makes it a group.
** On every infinite-dimensional topological vector space there is a discontinuous linear map.
** for every object
** for every object,
** The microcode can employ both pipelines to enable auto-repeating instructions such as rep movsw perform one iteration every clock cycle, while the 80486 needed three clocks per iteration ( and the earliest x86-chips significantly more than the 486 ).
** Jane was responsible for the elevator in every respect
** Online Variorum, showing every change between the six British editions.
** hypernyms: Y is a hypernym of X if every X is a ( kind of ) Y ( canine is a hypernym of dog )
** hyponyms: Y is a hyponym of X if every Y is a ( kind of ) X ( dog is a hyponym of canine )
** United Kingdom Census held, the first to record names and approximate ages of every household member and to be administered nationally.
** Circuit analysis, the process of finding the voltages across, and the currents through, every component in an electrical circuit
** Principal ideal domain, an integral domain in which every ideal is principal
** Unique factorization domain, an integral domain in which every non-zero element can be written as a product of irreducible elements in essentially a unique way
** Atomic domain, an integral domain in which every non-zero non-unit is a finite product of irreducible elements
** Dedekind domain, an integral domain in which every nonzero proper ideal factors into a product of prime ideals
** GCD domain, an integral domain in which every two non-zero elements have a greatest common divisor
** The Pythian Games ( founded 527 BC ) held in Delphi every four years
** The Nemean Games ( founded 516 BC ) held in Argolid every two years

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