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For example, the topological quotient of the metric space [,] identifying all points of the form (,) is not metrizable since it is not first-countable, but the quotient metric is a well-defined metric on the same set which induces a coarser topology. Moreover, different metrics on the original topological space (a disjoint union of countably ...
The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C ( a , b ) of continuous functions on ( a , b ) , for it may contain unbounded functions .
The space (0,1) ⊂ R, the open unit interval, is not a complete metric space with its usual metric inherited from R, but it is completely metrizable since it is homeomorphic to R. [6] The space Q of rational numbers with the subspace topology inherited from R is metrizable but not completely metrizable. [7]
A metric space is a length metric space if the intrinsic metric agrees with the original metric of the space. If the space has the stronger property that there always exists a path that achieves the infimum of length (a geodesic ) then it is called a geodesic metric space or geodesic space .
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 ...
In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.That is, a topological space (,) is said to be metrizable if there is a metric: [,) such that the topology induced by is . [1] [2] Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y. Metric convexity: does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers)
A compact metric space (X, d) also satisfies the following properties: Lebesgue's number lemma: For every open cover of X, there exists a number δ > 0 such that every subset of X of diameter < δ is contained in some member of the cover. (X, d) is second-countable, separable and Lindelöf – these three conditions are equivalent for metric ...