( Every datum is relatively oriented with respect to different portions of the geoid by the astro-geodetic methods already described.

Every year IMPETUS will have a theme and whole fest will be oriented in that direction.

Every set of points in the plane, and the lines connecting them, may be abstracted as the elements and flats of a rank-3 oriented matroid.

Every positive integer n > 1 can be represented in exactly one way as a product of prime powers:

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 completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

# Every prime ideal of A is principal.

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every third odd number is divisible by 3, which requires that no three successive odd numbers can be prime unless one of them is 3.

* Hardy and Littlewood listed as their Conjecture I: " Every large odd number ( n > 5 ) is the sum of a prime and the double of a prime.

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 strong probable prime to base a is also an Euler probable prime to the same base, but not vice versa.

Every closed 3-manifold has a prime decomposition: this means it is the connected sum of prime three-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).

Every non-zero characteristic is a prime number.

: Every Boolean algebra contains a prime ideal.

* Every number of the form 2 < sup > m </ sup > p for a natural number m and a prime number p such that p < 2 < sup > m + 1 </ sup > is also semiperfect.

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by

Iwasawa worked with so-called-extensions: infinite extensions of a number field with Galois group isomorphic to the additive group of p-adic integers for some prime p. Every closed subgroup of is of the form, so by Galois theory, a-extension is the same thing as a tower of fields such that.

Every imperfect field is necessarily transcendental over its prime subfield ( the minimal subfield ), because the latter is perfect.

A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.

* Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.

Every time I closed my eyes, I saw Gray Eyes rushing at me with a knife.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

* Every closed subgroup of a profinite group is itself profinite ; the topology arising from the profiniteness agrees with the subspace topology.

Every open subgroup H is also closed, since the complement of H is the open set given by the union of open sets gH for g in G

* Every closed nowhere dense set is the boundary of an open set.

Every map that is injective, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed.

Every base he closed resulted in a new construction project elsewhere to expand another base, relocation of forces projects and other related spending.

Every year the central business district ( with corners at the Municipal Building, Grand Street Fire House and Croton-Harmon High School ) is closed to automobile traffic for music, American food, local fund raisers, traveling, and local artists.

The generalized Poincaré conjecture states that Every simply connected, closed n-manifold is homeomorphic to the n-sphere.

Every closed subspace of a reflexive space is reflexive.

* Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

Every October the high street is closed for the two Saturdays either side of 11 October for the Marlborough Mop Fair.

Every closed point of Hilb ( X ) corresponds to a closed subscheme of a fixed scheme X, and every closed subscheme is represented by such a point.

Every homeomorphism is open, closed, and continuous.

Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,