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Page "Axiom of choice" ¶ 96
from Wikipedia ## 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 ** 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, ** 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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