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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 ).

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## Some Related Sentences

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__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__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__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 3-manifold

__Every__

__3-manifold__

**has**many embedded 2-spheres, and

**a**2-sphere embedded

**in**

**a**

__3-manifold__never

**has**

**a**compressing disc.

__Every__

**closed**, orientable,

**connected**

__3-manifold__

**is**obtained by performing Dehn surgery on

**a**link

**in**

**the**3-sphere.

Every and has

__Every__woman

__has__had

**the**experience

**of**saying no when she meant yes, and saying yes when she meant no.

__Every__detail

**in**his interpretation

__has__been beautifully thought out, and

**of**these I would especially cite

**the**delicious laendler touch

**the**pianist brings to

**the**fifth variation

**(**an obvious indication that he

**is**playing with Viennese musicians ), and

**the**gossamer shading throughout.

__Every__family

**of**Riviera Presbyterian Church

__has__been asked to read

**the**Bible and pray together daily during National Christian Family Week and to undertake one project

**in**which all members

**of**

**the**family participate.

__Every__community, if

**it**

**is**alive

__has__

**a**spirit, and that spirit

**is**

**the**center

**of**its unity and identity.

__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.
** Tukey's lemma

**:**__Every__non-empty collection**of**finite character__has__**a**maximal element with respect to inclusion.
*

__Every__continuous functor on**a**small-complete category which satisfies**the**appropriate solution set condition__has__**a**left-adjoint**(****the**Freyd adjoint functor theorem**).**__Every__unit

**of**length

__has__

**a**corresponding unit

**of**area, namely

**the**area

**of**

**a**square with

**the**given side length.

__Every__ATM cell

__has__an 8-or 12-bit Virtual Path Identifier

**(**VPI ) and 16-bit Virtual Channel Identifier

**(**VCI ) pair defined

**in**its header.

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