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The unit interval is a complete metric space, homeomorphic to the extended real number line.
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(BCT1) Every complete metric space is a Baire space.
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Let E be a complete metric space and f a contraction mapping of E into itself.
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BCT1 also shows that every complete metric space with no isolated points is uncountable.
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Every Gδ-subset of the Hilbert cube is a Polish space, a topological space homeomorphic to a separable and complete metric space.
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(BCT3) A non-empty complete metric space, or any of its subsets with nonempty interior, is not the countable union of nowhere-dense sets.
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Also: if a non-empty complete metric space is the countable union of closed sets, then one of these closed sets has non-empty interior.
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Equivalently, it is a complete separable metric space whose metric has been "forgotten".
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If such a space is complete (as a metric space) then it is called a Banach space.
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In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces.
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Moreover, the Banach fixed-point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point.
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Note that neither of these statements implies the other, since there are complete metric spaces which are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces which are not metrizable (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in Functional Analysis; the uncountable Fort space).
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Equivalence between the two types of randomization is proven in the case the set of actions is a complete separable metric space, which is the case considered by Wald and Wolfowitz for sequential decision problem.
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They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.
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A Polish space is a second-countable topological space that is metrizable with a complete metric.
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Other examples include: Every metric space is Tychonoff; every pseudometric space is completely regular.
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Complete metric space - Wikipedia, the free encyclopedia
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In this way, we obtain a complete metric space which is also a field and contains Q.
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(Redirected from Completion (metric space))
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Two of the main results are the following: • Theorem: Let X and Y be complete metric spaces.
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Complete metric space - Wikipedia
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(BCT1) Every complete metric space is a Baire space.
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We have seen that the Banach fixed point theorem applies to contractions on complete metric spaces.
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The metric expansion of space is of a kind completely different from the expansions and explosions seen in daily life.
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Generalising both the metric spaces and the topological groups, every uniform space is completely regular.
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