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* Every subset of Baire space or Cantor space is an open set in the usual topology on the space.

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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 Baire

*****

__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__Polish

**space**

**is**obtained as a continuous image

**of**

__Baire__

**space**;

**in**fact every Polish

**space**

**is**

**the**image

**of**a continuous bijection defined

**on**a closed

**subset**

**of**

__Baire__

**space**

**.**

*****

__Every__

**set**

**of**reals

**in**L ( R )

**is**Lebesgue measurable (

**in**fact, universally measurable ) and has

**the**property

**of**

__Baire__and

**the**perfect

**set**property

**.**

Every and space

__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__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__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__node

**on**

**the**Freenet network contributes storage

__space__to hold files, and bandwidth that it uses to route requests from its peers

**.**

*****

__Every__Lie group

**is**parallelizable, and hence

**an**orientable manifold ( there

**is**a bundle isomorphism between its tangent bundle and

**the**product

**of**itself with

**the**tangent

__space__at

**the**identity )

__Every__vector

__space__has a basis, and all bases

**of**a vector

__space__have

**the**same number

**of**elements, called

**the**dimension

**of**

**the**vector

__space__

**.**

__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

**.**

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