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Completeness is a property of the metric and not of the topology, meaning that a complete metric space can be homeomorphic to a non-complete one. An example is given by the real numbers, which are complete but homeomorphic to the open interval (0,1) , which is not complete.
In fact, every metric space has a unique completion, which is a complete space that contains the given space as a dense subset. For example, [0, 1] is the completion of (0, 1) , and the real numbers are the completion of the rationals.
The distinction between a completely metrizable space and a complete metric space lies in the words there exists at least one metric in the definition of completely metrizable space, which is not the same as there is given a metric (the latter would yield the definition of complete metric space). Once we make the choice of the metric on a ...
In mathematics, more specifically in functional analysis, a Banach space (/ ˈ b ɑː. n ʌ x /, Polish pronunciation:) is a complete normed vector space.Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is ...
Let T : X → X be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are: Assume that some iterate T n of T is a contraction. Then T has a unique fixed point. Assume that for each n, there exist c n such that d(T n (x), T n (y)) ≤ c n d(x, y) for all x and y, and that
In mathematics, a topological space is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. [1] According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are examples of Baire spaces.
Given a (possibly incomplete) measure space (X, Σ, μ), there is an extension (X, Σ 0, μ 0) of this measure space that is complete. [3] The smallest such extension (i.e. the smallest σ-algebra Σ 0) is called the completion of the measure space. The completion can be constructed as follows:
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number < such that for all x and y in M,