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Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.

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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**

**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 nowhere

" Or

**,**as Rainer Maria Rilke puts it**,**"__Every__artist**is**born in an alien country ; he has**a**homeland__nowhere__but within his own borders**.**

Every and dense

__Every__morning early

**,**in

**the**summer

**,**we searched

**the**trunks

**of**

**the**trees as high as we could reach for

**the**locust shells

**,**carefully detached their hooked claws from

**the**bark where they hung

**,**

**and**stabled them

**,**

**a**weird faery herd

**,**in an angle between

**the**high roots

**of**

**the**tulip tree

**,**where no grass grew in

**the**

__dense__shade

**.**

*

__Every__topological space X**is****a**__dense__subspace**of****a**compact space having at most one point more than X**,**by**the**Alexandroff one-point compactification**.**__Every__normed vector space V sits as

**a**

__dense__subspace inside

**a**Banach space ; this Banach space

**is**essentially uniquely defined by V

**and**

**is**called

**the**completion

**of**V

**.**

__Every__evening they fly

**,**often in groups

**and**sometimes over long distances

**,**to reach safe roosting sites such as

__dense__trees or shrubs that impede predator movement

**,**or

**,**at higher latitudes

**,**

__dense__conifers that afford good wind protection

**.**

*

__Every__non-empty Baire space**is****of**second category in itself**,****and**every intersection**of**countably**many**__dense__open subsets**of**X**is**non-empty**,**but**the**converse**of**neither**of**these**is**true**,**as**is**shown by**the**topological disjoint sum**of****the**rationals**and****the**unit interval 1**.**__Every__sequence

**of**A

**,**B

**,**

**and**C without immediate repetition

**of**

**the**same one

**is**possible

**and**gives an equally

__dense__packing for spheres

**of**

**a**given radius

**.**

__Every__autumn

**,**

**the**dried needles

**of**this tree forms

**a**

__dense__carpet on

**the**forest floor

**,**which

**the**locals gather in large bundles to serve as bedding for their cattle

**,**for

**the**year round

**.**

Every and set

*

__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__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__corporation

**,**whether financial or

**union**

**,**as well as every division

**of**

**the**administration

**,**were

__set__up as branches

**of**

**the**party

**,**

**the**CEOs

**,**Union leaders

**,**

**and**division directors being sworn-in as section presidents

**of**

**the**party

**.**

__Every__DNS zone must be assigned

**a**

__set__

**of**authoritative name servers that are installed in NS records in

**the**parent zone

**,**

**and**should be installed ( to be authoritative records ) as self-referential NS records on

**the**authoritative name servers

**.**

Group actions / representations:

__Every__group G can be considered as**a**category with**a**single object whose morphisms are**the**elements**of**G**.**A functor from G to Set**is**then nothing but**a**group action**of**G on**a**particular__set__**,**i**.**e**.****a**G-set**.**
# " Personality " Argument: this argument

**is**based on**a**quote from Hegel: "__Every__man has**the**right to turn his will upon**a**thing or make**the**thing an object**of**his will**,**that**is**to say**,**to__set__aside**the**mere thing**and**recreate it as his own ".__Every__atom across this plane has an individual

__set__

**of**emission cones .</ p > < p > Drawing

**the**billions

**of**overlapping cones

**is**impossible

**,**so this

**is**

**a**simplified diagram showing

**the**extents

**of**all

**the**emission cones combined

**.**

*

__Every__preorder can be given**a**topology**,****the**Alexandrov topology ;**and**indeed**,**every preorder on**a**__set__**is**in one-to-one correspondence with an Alexandrov topology on that__set__**.**__Every__binary relation R on

**a**

__set__S can be extended to

**a**preorder on S by taking

**the**transitive closure

**and**reflexive closure

**,**R < sup >+=</ sup >.

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