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

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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__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 bounded

*

__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__totally ordered set**that**is a__bounded__lattice is also a Heyting algebra, where is equal to when, and 1 otherwise**.**
This is really a special case

**of**a more general fact:__Every__continuous function from a compact space into a metric space is__bounded__**.**
For example, to study

**the**theorem “__Every____bounded__sequence**of****real****numbers**has a supremum ” it is necessary to use a base system which can speak**of****real****numbers**and sequences**of****real****numbers****.**__Every__maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every

__bounded__face

**of**a maximal outerplanar graph is a triangle

**.**

__Every__three years

**the**Company makes

**an**award to

**the**three buildings

**or**structures, in

**the**area

__bounded__by

**the**M25 motorway, which respectively embody

**the**most outstanding example

**of**brickwork,

**of**slated

**or**tiled roof and

**of**hard-surface tiled wall and /

**or**floor

**.**

__Every__Polish town was

__bounded__to put up a quantity

**of**soldiers-this was a conspicuous sign

**of**a power

**of**a given town how much soldiers it had to put up

**.**

__Every__complete lattice is also a

__bounded__lattice, which is to say

**that**it has a greatest and

**least**element

**.**

__Every__

__bounded__linear transformation from a normed vector space to a complete, normed vector space can be uniquely extended to a

__bounded__linear transformation from

**the**completion

**of**to

**.**

__Every__

__bounded__positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0

**.**improved this criterion by showing

**that**it is sufficient to ask

**that**, for every continuous positive-definite compactly supported function f on G,

**the**function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes

**the**modular function

**.**

__Every__subset

**of**a totally

__bounded__space is a totally

__bounded__set ; but even if a space is not totally

__bounded__, some

**of**its subsets still will be

**.**

__Every__topological vector space X gives a bornology on X by defining a subset to be

__bounded__iff for all open sets containing zero there exists a with

**.**

__Every__(

__bounded__) convex polytope is

**the**image

**of**a simplex, as every

**point**is a convex combination

**of**

**the**( finitely many ) vertices

**.**

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