 Page "Separable space" ¶ 36
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Every and separable * Every compact metric space is separable. * Every compact metric space ( or metrizable space ) is separable. * 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 second-countable space is first-countable, separable, and Lindelöf. * Every subextension of F / k is separable. * Every finite subextension of F / k is separable. * Every polynomial over k is separable. * Every finite extension of k is separable. * Every algebraic extension of k is separable. * Every reduced commutative k-algebra A is a separable algebra ; i. e., is reduced for every field extension F / k. Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L < sup >∞</ sup >( X ) for some standard measure space ( X, μ ) and conversely, for every standard measure space X, L < sup >∞</ sup >( X ) is a von Neumann algebra. Every separable topological space is ccc. Every metric space which is ccc is also separable, but in general a ccc topological space need not be separable. Every second-countable manifold is separable and paracompact.

Every and metric Every compact metric space is complete, though complete spaces need not be compact. * Every metric space is Tychonoff ; every pseudometric space is completely regular. Every uniformly continuous function between metric spaces is continuous. *( BCT1 ) Every complete metric space is a Baire space. Every smooth submanifold of R < sup > n </ sup > has an induced Riemannian metric g: the inner product on each tangent space is the restriction of the inner product on R < sup > n </ sup >. *( BCT1 ) Every complete metric space is a Baire space. Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space. * Every totally disconnected compact metric space is homeomorphic to a subset of a countable product of discrete spaces. This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded. # Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 ( real ) dimensions. Every hyperkähler manifold M has a 2-sphere of complex structures ( i. e. integrable almost complex structures ) with respect to which the metric is Kähler. Every special uniformly continuous real-valued function defined on the metric space is uniformly approximable by means of Lipschitz functions. * Every totally bounded metric space is bounded. Every compact metric space is totally bounded. Every locally compact group which is second-countable is metrizable as a topological group ( i. e. can be given a left-invariant metric compatible with the topology ) and complete. Every ( almost ) complex manifold admits a Hermitian metric.

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 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 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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