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A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov–Hausdorff ...
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
This implies e. g. that every completely metrizable topological vector space is complete. Indeed, a topological vector space is called complete iff its uniformity (induced by its topology and addition operation) is complete; the uniformity induced by a translation-invariant metric that induces the topology coincides with the original uniformity.
Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological ...
Distances between points are defined in a metric space. Isomorphisms between metric spaces are called isometries. Every metric space is also a topological space. A topological space is called metrizable, if it underlies a metric space. All manifolds are metrizable. In a metric space, we can define bounded sets and Cauchy sequences. A metric ...
A set with a topology is called a topological space. Metric spaces are an important class of topological spaces where a real, non-negative distance, also called a metric, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.
A topological space is said to be a pseudometrizable space [4] if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space. The difference between pseudometrics and metrics is entirely topological.
This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists a real number >, such that is in whenever the distance (,) <. This definition generalizes to topological spaces by replacing "open ball" with " open set ".