** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

# Every open cover of A has a finite subcover.

* Countably compact: Every countable open cover has a finite subcover.

# Every finite and contingent being has a cause.

Every finite simple group is isomorphic to one of the following groups:

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

Every finite tree structure has a member that has no superior.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

* Every finite tree with n vertices, with, has at least two terminal vertices ( leaves ).

Every finite group of exponent n with m generators is a homomorphic image of B < sub > 0 </ sub >( m, n ).

Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0.

Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.

Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index.

Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero.

Every finite group has a composition series, but not every infinite group has one.

* Every finite or cofinite subset of the natural numbers is computable.

* Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to

Every finite ordinal ( natural number ) is initial, but most infinite ordinals are not initial.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

* Every commutative semisimple ring must be a finite direct product of fields.

* Every subextension of F / k is separable.

Every adjunction 〈 F, G, ε, η 〉 extends an equivalence of certain subcategories.

Every adjunction 〈 F, G, ε, η 〉 gives rise to an associated monad 〈 T, η, μ 〉 in the category D. The functor

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).

Every statement in F #, including statements and loops, is a composable expression with a definite return type as well.

Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.

* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.

Every functor F: D → E induces a functor F < sup > C </ sup >: D < sup > C </ sup > → E < sup > C </ sup > ( by composition with F ).

* Every reduced commutative k-algebra A is a separable algebra ; i. e., is reduced for every field extension F / k.

President John F. Kennedy said of Hayden, " Every Federal program which has contributed to the development of the West — irrigation, power, reclamation — bears his mark, and the great Federal highway program which binds this country, together, which permits this State to be competitive east and west, north and south, this in large measure is his creation.

* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.

Every President of the United States from William Howard Taft to John F. Kennedy threw a ceremonial first pitch at least once at Griffith.

* " Every Time You Say I Love You " ( 1981 ) F & SF, May

Every sitting U. S. President since 1960 ( Eisenhower, John F. Kennedy, Lyndon B. Johnson, Richard Nixon, Gerald Ford, Jimmy Carter, Ronald Reagan, George H. W. Bush, Bill Clinton and George W. Bush ) has addressed a convention or attended an event at Cobo Center.

" Two Cars in Every Garage and Three Eyes on Every Fish " was the first episode produced for the second season, but " Bart Gets an F " aired first because of Bart's popularity and the producers wanted to premiere with an episode involving him in hopes of stealing viewers from The Cosby Show.

* Young, F. Every Man his own Mechanic ( 1881 )

* The New Dictionary of Cultural Literacy, Third Edition: What Every American Needs to Know Edited by James Trefil, Joseph F. Kett, and E. D. Hirsch.

" Every time I turn around, there's a new trial of the century ," says defense attorney F. Lee Bailey.

Every year since 1952, it has sponsored the Alfred Korzybski Memorial Lecture, with presenters from a broad range of disciplines, from science to medicine to entertainment, including names like actor Steve Allen, psychologist Albert Ellis, scientist and visionary R. Buckminster Fuller, linguist Allen Walker Read, and philosopher F. S. C. Northrop.