[permalink] [id link]

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

from
Wikipedia

## Some Related Sentences

Every and cofinal

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 partially

Most

**of**his films have been at least__partially__financed by Telefilm Canada, and Cronenberg is**a**vocal supporter**of**government-backed film projects, saying "__Every__country needs system**of**government Grant ( money ) | grants in order to have**a**national cinema in the face**of**Hollywood ".*****

__Every__non-empty

**set**

**of**left ideals

**of**R,

__partially__

**ordered**by inclusion, has

**a**

**maximal**element with respect to

**set**inclusion

**.**

Her recording debut was actually made in 1978 when she sang back-up ( and

__partially__lead ) vocals for The Michael Zager Band's " Life's**a**Party " and background vocals on Chaka Khan's hit single " I'm__Every__Woman "—**a**song she would turn into**a**larger hit for herself in 1993**.**

Every and ordered

__Every__

__ordered__field is

**a**formally real field, i

**.**e., 0 cannot be written as

**a**sum

**of**nonzero squares

**.**

*****

__Every__totally

__ordered__

**set**

**that**is

**a**bounded lattice is also

**a**Heyting algebra, where is equal to when, and 1 otherwise

**.**

__Every__time she entered, song typical

**of**the Brazilian northeast would play, stopping only when

__ordered__by Cirene herself

**.**

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 >.

0.616 seconds.