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* Every finite or cofinite subset of the natural numbers is computable.
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Every and finite
** 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 subset of may be covered by a finite set of positive orthants, whose apexes all belong to
* Every finite subextension of F / k is separable.
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 and cofinite
* Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.
Every and subset
Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.
** 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.
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
** Every infinite game in which is a Borel subset of Baire space is determined.
# Every infinite subset of X has a complete accumulation point.
# Every infinite subset of A has at least one limit point in A.
* Limit point compact: Every infinite subset has an accumulation point.
Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.
* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.
* Every separable metric space is homeomorphic to a subset of the Hilbert cube.
* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.
* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.
* Every separable metric space is isometric to a subset of the
Every element s, except a possible greatest element, has a unique successor ( next element ), namely the least element of the subset of all elements greater than s. Every subset which has an upper bound has a least upper bound.
Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.
* Every subset of Baire space or Cantor space is an open set in the usual topology on the space.
* Every arithmetical subset of Cantor space of < sup >( or?
Every subset of the Hilbert cube inherits from the Hilbert cube the properties of being both metrizable ( and therefore T4 ) and second countable.
It is more interesting that the converse also holds: Every second countable T4 space is homeomorphic to a subset of the Hilbert cube.
* Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces.
* 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.
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