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* Every nonempty affine algebraic set may be written uniquely as a union of algebraic varieties ( where none of the sets in the decomposition are subsets of each other ).
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Every and nonempty
: Every nonempty internal set that has an internal upper bound has a least internal upper bound.
: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.
Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X.
Scarf ’ s Theorem: Every balanced game has a nonempty core.
Every and affine
Every smooth surface S has a unique affine plane tangent to it at each point.
Every object in the drawing can be subjected to arbitrary affine transformations: moving, rotating, scaling, skewing and a configurable matrix.
( 5 ) Every continuous affine isometric action of G on a real Hilbert space has a fixed point ( property ( FH )).
Every affine vector field is a curvature collineation.
Every affine vector field is a curvature collineation.
Every and algebraic
** Every field has an algebraic closure.
Every root of a polynomial equation whose coefficients are algebraic numbers is again algebraic.
Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.
Every epimorphism in this algebraic sense is an epimorphism in the sense of category theory, but the converse is not true in all categories.
* Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer.
Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation ( s ) defining the structure.
Every algebraic curve C of genus g ≥ 1 is associated with an abelian variety J of dimension g, by means of an analytic map of C into J.
Every field has an algebraic closure, and it is unique up to an isomorphism that fixes F.
* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.
* Every ( biregular ) algebraic automorphism of a projective space is projective linear.
* Every algebraic extension of k is separable.
* Every real algebraic number field K of degree n contains a PV number of degree n. This number is a field generator.
* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.
Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub ( A ) for some algebra A.
* Every character value is a sum of n m < sup > th </ sup > roots of unity, where n is the degree ( that is, the dimension of the associated vector space ) of the representation with character χ and m is the order of g. In particular, when F is the field of complex numbers, every such character value is an algebraic integer.
* Every finite poset is directed complete and algebraic.
* Let K < sup > a </ sup > be an algebraic closure of K containing L. Every embedding σ of L in K < sup > a </ sup > which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.
Every algebraic plane curve has a degree, which can be defined, in case of an algebraically closed field, as number of intersections of the curve with a generic line.
Every planar graph has an algebraic dual, which is in general not unique ( any dual defined by a plane embedding will do ).
Every and set
: Every set has a choice function.
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.
** Well-ordering theorem: Every set can be well-ordered.
** 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.
** 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 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 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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