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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 .
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 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 ...
Complete metric space – Metric geometry; Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point; Completely uniformizable space; Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
A closed subspace of a Baire space need not be Baire. See the Examples section. If a space contains a dense subspace that is Baire, it is also a Baire space. [17] A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space. [18] [19] Every topological sum of Baire spaces is Baire. [20]
CAT(k) space; Category of metric spaces; Cauchy sequence; Cayley–Klein metric; Chebyshev distance; Chow–Rashevskii theorem; Classical Wiener space; Clifton–Pohl torus; Coarse structure; Comparison triangle; Complete metric space; Conformal dimension; Contraction mapping; Convex cap; Convex metric space; Covering number; Curve
A metric space is called complete if all Cauchy sequences converge. Every incomplete space is isometrically embedded, as a dense subset, into a complete space (the completion). Every compact metric space is complete; the real line is non-compact but complete; the open interval (0,1) is incomplete. Every Euclidean space is also a complete metric ...