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* 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.

from
Wikipedia

## Some Related Sentences

Every and non-empty

** 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**.**
** Tukey's lemma:

__Every____non-empty__collection**of**finite character has a maximal element with respect to inclusion**.**__Every__

__non-empty__

**intersection**

**of**a 3-sphere with a three-dimensional hyperplane

**is**a 2-sphere ( unless

**the**hyperplane

**is**tangent to

**the**3-sphere

**,**

**in**which case

**the**

**intersection**

**is**a single point ).

*****

__Every__

__non-empty__set

**of**left ideals

**of**R

**,**partially ordered

**by**inclusion

**,**has a maximal element with respect to set inclusion

**.**

__Every__

__non-empty__file must have at least one fork

**,**

**and**depending on

**the**file system

**,**a file may have one or more other associated forks

**,**which

**in**turn may contain primary data integral to

**the**file

**,**or just metadata

**.**

Every and Baire

*****

__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__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__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

**.**

Every and is

__Every__legislator from Brasstown Bald to Folkston

__is__going to have his

**every**vote subjected to

**the**closest scrutiny

**as**a test

**of**his political allegiances

**,**not his convictions

**.**

__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__taxpayer

__is__well aware

**of**

**the**vast size

**of**our annual defense budget

**and**most

**of**our readers also realize that a large portion

**of**

**these**expenditures go for military electronics

**.**

__Every__few days

**,**

**in**

**the**early morning

**,**

**as**

**the**work progressed

**,**twenty men would appear to push it ahead

**and**to shift

**the**plank foundation that distributed its weight widely on

**the**Rotunda pavement

**,**supported

**as**it

__is__

**by**ancient brick vaulting

**.**

__Every__dream

**,**

**and**this

__is__

**true**

**of**a mental image

**of**any type even though it may be readily interpreted into its equivalent

**of**wakeful thought

**,**

__is__a psychic phenomenon for which no explanation

__is__available

**.**

__Every__library borrower

**,**or at least those whose taste goes beyond

**the**five-cent fiction rentals

**,**knows what it

__is__to hear

**the**librarian say apologetically

**,**`` I'm sorry

**,**

**but**we don't have that book

**.**

__Every__community

**,**if it

__is__alive has a spirit

**,**

**and**that spirit

__is__

**the**center

**of**its unity

**and**identity

**.**

The restricted principle "

__Every__partially ordered set has a maximal totally ordered subset "__is__also equivalent to AC over ZF**.**__Every__natural-born citizen

**of**a foreign state who

__is__also an American citizen

**and**

**every**natural-born American citizen who

__is__a citizen

**of**a foreign land owes a double allegiance

**,**one to

**the**United States

**,**

**and**one to his homeland (

**in**

**the**event

**of**an immigrant becoming a citizen

**of**

**the**US ), or to his adopted land (

**in**

**the**event

**of**an emigrant natural born citizen

**of**

**the**US becoming a citizen

**of**another nation ).

__Every__line

**of**written text

__is__a mere reflection

**of**references from any

**of**a multitude

**of**traditions

**,**or

**,**

**as**Barthes puts it

**,**"

**the**text

__is__a tissue

**of**quotations drawn from

**the**innumerable centres

**of**culture "; it

__is__never original

**.**

__Every__year

**,**on

**the**last Sunday

**in**April

**,**there

__is__an ice fishing competition

**in**

**the**frozen estuarine waters

**of**

**the**Anadyr River's mouth

**.**

__Every__lattice element

**of**

**the**structure

__is__

**in**its proper place

**,**whether it

__is__a single atom or a molecular grouping

**.**

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