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* Every ( biregular ) algebraic automorphism of a projective space is projective linear.
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Every and biregular
Every edge-transitive graph must be bipartite and either semi-symmetric or biregular.
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 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 ).
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 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 automorphism
Every inner automorphism is indeed an automorphism of the group G, i. e. it is a bijective map from G to G and it is a homomorphism ; meaning ( xy )< sup > a </ sup >
Every non-inner automorphism yields a non-trivial element of Out ( G ), but different non-inner automorphisms may yield the same element of Out ( G ).
Every homomorphism of the Petersen graph to itself that doesn't identify adjacent vertices is an automorphism.
* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).
Every and projective
Every projective variety is complete, but not vice versa.
* Every holomorphic line bundle on a projective variety is a line bundle of a divisor.
* Every module over a field or skew field is projective ( even free ).
* Every projective module is flat.
Every module possesses a projective resolution.
Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve.
Every rational variety, including the projective spaces, is rationally connected, but the converse is false.
Every compact, connected 2-manifold ( or surface ) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes.
* Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element of L ( R ).
Every and space
** Every vector space has a basis.
** Every infinite game in which is a Borel subset of Baire space is determined.
** Every Tychonoff space has a Stone – Čech compactification.
* Theorem Every reflexive normed space is a Banach 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 compact metric space is separable.
* 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.
* Pseudocompact: Every real-valued continuous function on the space is bounded.
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 compact metric space is complete, though complete spaces need not be compact.
Every point in three-dimensional Euclidean space is determined by three coordinates.
Every node on the Freenet network contributes storage space to hold files, and bandwidth that it uses to route requests from its peers.
Every space filling curve hits some points multiple times, and does not have a continuous inverse.
* 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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