 Page "Axiom of choice" ¶ 49
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** and theorem ** 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 theorem that every Hilbert space has an orthonormal basis. ** 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 ** Every surjective function has a right inverse. ** 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. ** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion. ** Antichain principle: Every partially ordered set has a maximal antichain. ** Every vector space has a basis. ** Every unital ring other than the trivial ring contains a maximal ideal. ** Every infinite game in which is a Borel subset of Baire space is determined. ** Every field has an algebraic closure. ** Every field extension has a transcendence basis. ** Every Tychonoff space has a Stone – Čech compactification. ** 2 ( 2012 ) ( Every hospital, airport and military base has Helipads. ** It Happens Every Spring-Valentine Davies, Shirley Smith ** Every voiced consonant is written with a deep letter similar in shape to the corresponding voiceless consonant which in turn is written with a tall letter. ** The Fugitive Guy: Every so often, Letterman would introduce " Roger Campbell " ( Elliott, wearing an extremely bad toupée ), a new member of the Late Night crew. ** Edwin Hawkins for " Every Man Wants To Be Free " performed by the Edwin Hawkins Singers ** The Police for " Every Breath You Take " ** Luther Vandross and Marcus Miller for " Little Miracles ( Happen Every Day )" performed by Luther Vandross ** L. A. Reid and Babyface for " Every Little Step " performed by Bobby Brown ** 1908 Feb, in Every Where Vol. 21 ** A Journey Through Every Stage of Life ** 1664-The Concession and Agreement of the Lords Proprietors of the Province of New Caesarea, or New Jersey, to and With All and Every the Adventurers and All Such as Shall Settle or Plant There ** Sting ( songwriter ) for " Every Breath You Take " performed by The Police ** The Police for " Every Breath You Take " ** Every item entry must be within one level of its predecessor, such that each item must be a sibling or child of the preceding item ( thus, no item can be a grandchild of the preceding item ).

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