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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 infinite game in which is a Borel subset of Baire space is determined.
# Every infinite subset of X has a complete accumulation point.
# Every infinite subset of A has at least one limit point in A.
* Limit point compact: Every infinite subset has an accumulation point.
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 cofinal subset of a partially ordered set must contain all maximal elements of that set.
* Every separable metric space is homeomorphic to a subset of the Hilbert cube.
* 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 separable metric space is isometric to a subset of the
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 Baire space or Cantor space is an open set in the usual topology on the space.
* Every arithmetical subset of Cantor space of < sup >( or?
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 finite or cofinite subset of the natural numbers is computable.

Every and nowhere
* Every closed nowhere dense set is the boundary of an open set.
" 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 intersection of countably many dense open sets is dense.
* 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 set has a choice function.
** Well-ordering theorem: Every set can be well-ordered.
** Antichain principle: Every partially ordered set has a maximal antichain.
* 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 non-empty set A contains an element B which is disjoint from A.
* 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 singleton set
Every processor or processor family has its own machine code instruction set.
Every set is a class, no matter which foundation is chosen.
Every non-empty totally ordered set is directed.
* 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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