 Page "Convex set" ¶ 19
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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 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 a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense. * 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 vector ** Every vector space has a basis. Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism. 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 finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ). Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. Every vector in the space may be written as a linear combination of unit vectors. Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space. Every coalgebra, by ( vector space ) duality, gives rise to an algebra, but not in general the other way. * Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle. For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ". Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted. Every vector space is free, and the free vector space on a set is a special case of a free module on a set.

Every and space ** 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 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 )

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