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* Every subset of a complete Boolean algebra has a supremum, by definition ; it follows that every subset also has an infimum ( greatest lower bound ).

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## Some Related Sentences

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.

Every and complete

__Every__Hilbert space X is

**a**Banach space because

**,**

**by**

**definition**

**,**

**a**Hilbert space is

__complete__with respect to the norm associated with its inner product

**,**where

**a**norm and

**an**inner product are said to be associated if for all x ∈ X.

__Every__soul

**has**to follow the path

**,**as explained

**by**the Jinas and revived

**by**the tirthankaras

**,**to attain

__complete__liberation or nirvana.

__Every__time Emperor Wen sent generals out on battles

**,**he required them to follow the

__complete__battle plans

**that**he had drafted

**,**and even the dates for battles needed approval from the emperor.

__Every__use

**of**

**a**bank machine

**,**payment

**by**credit card

**,**use

**of**

**a**phone card

**,**call from home

**,**checked out library book

**,**rented video

**,**or otherwise

__complete__recorded transaction generates

**an**electronic record.

__Every__time he is thrown out

**of**the house

**,**he is shown wearing the same shirt although he does not always wear

**it**when he is thrown out

**(**the producers never shot

**a**second sequence with Jazz being thrown out

**of**the house

**,**only adjusting the original scene for time purposes

**;**

**an**exception is in the episode " Community Action ", where Jazz was thrown out along with

**a**lifesize cardboard cut-out

**of**Bill Cosby

**,**

__complete__with

**a**blooper showing Jeff Townes reshooting his flying off the house several times

**).**

__Every__two years

**,**before migration

**,**the adult Common Crane undergoes

**a**

__complete__moult

**,**remaining flightless for six weeks

**,**until the new feathers grow.

"

__Every__opening phase**of****a**Canadian operation was**a**__complete__success and the staff works**a**mathematical masterpiece … the Canadian Army never followed up their opening successes to reach**a**__complete__victory.__Every__

**Boolean**

**algebra**is

**a**Heyting

**algebra**when

**a**→ b is defined as usual as ¬

**a**∨ b

**,**as is

**every**

__complete__distributive lattice when

**a**→ b is taken to be the

**supremum**

**of**the set

**of**all c for which

**a**∧ c ≤ b. The open sets

**of**

**a**topological space form

**a**

__complete__distributive lattice and hence

**a**Heyting

**algebra**.

__Every__5 years

**,**each Commission conducts

**a**

__complete__review

**of**all constituencies in its part

**of**the United Kingdom.

__Every__month

**,**

**a**employer must file

**a**

__complete__IR348 Employer monthly schedule with the IRD

**,**stating the income and deductions

**of**each employee.

__Every__

__complete__lattice is

**also**

**a**bounded lattice

**,**which is to say

**that**

**it**

**has**

**a**

**greatest**and least element.

__Every__Skolem theory is model

__complete__

**,**i. e.

**every**substructure

**of**

**a**model is

**an**elementary substructure.

__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__year on the third Saturday in September

**,**German-Americans celebrate the Annual Steuben Parade on Fifth Avenue and

**an**Oktoberfest-style beer fest

__complete__with food and live music in Central Park.

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