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

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## Some Related Sentences

Every and finite

** Tukey's lemma:

__Every__non-empty collection**of**__finite__character has a maximal element with respect to inclusion**.**
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__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__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-dimensional central simple algebra over a

__finite__field must be a matrix ring over that field

**.**

Every and cofinite

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__

__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__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__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**

**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

**.**

0.213 seconds.