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A metric space M is compact if every open cover has a finite subcover (the usual topological definition). A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.) A metric space M is compact if it is complete and ...
The space of real numbers and the space of complex numbers (with the metric given by the absolute difference) are complete, and so is Euclidean space, with the usual distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces .
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 ...
The product of a finite set of metric spaces in Met is a metric space that has the cartesian product of the spaces as its points; the distance in the product space is given by the supremum of the distances in the base spaces. That is, it is the product metric with the sup norm. However, the product of an infinite set of metric spaces may not ...
If X is a separable, complete metric space with no isolated points, the cardinality of X is exactly . If X is a locally compact Hausdorff space with no isolated points, there is an injective function (not necessarily continuous) from Cantor space to X , and so X has cardinality at least 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} .
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, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
A notable example of this construction is the Seifert–Weber space which is obtained by gluing opposite faces of a regular dodecahedron. A variation on this construction is by using hyperbolic Coxeter polytopes (polytopes whose dihedral angles are of the form π / m , m ∈ N {\displaystyle \pi /m,m\in \mathbb {N} } ).