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* 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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## Some Related Sentences

Every and irreducible

__Every__

**polynomial**in can be factorized into polynomials that are

__irreducible__over F

**.**This factorization

**is**unique up to permutation

**of**

**the**factors and

**the**multiplication

**of**

**the**factors by nonzero constants from F ( because

**the**ring

**of**polynomials over

**a**field

**is**

**a**unique factorization domain whose units are

**the**nonzero constant polynomials ).

__Every__Heyting algebra with exactly

**one**coatom

**is**subdirectly

__irreducible__, whence every Heyting algebra can be made an SI by adjoining

**a**new top

**.**

Every and closed

__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__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__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__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__integrable subbundle

**of**

**the**tangent bundle — that

**is**,

**one**whose sections are

__closed__under

**the**Lie bracket — also defines

**a**Lie algebroid

**.**

__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__

__closed__curve c on X

**is**homologous to for

**some**simple

__closed__curves c

**<**sub

**>**

**i**

**</**sub >, that

**is**,

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

**.**

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