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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 ...
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces , and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.
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.
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 ...
A pointed metric space is a pair (X,p) consisting of a metric space X and point p in X. A sequence (X n, p n) of pointed metric spaces converges to a pointed metric space (Y, p) if, for each R > 0, the sequence of closed R-balls around p n in X n converges to the closed R-ball around p in Y in the usual Gromov–Hausdorff sense. [10]
Download as PDF; Printable version ... is a metric space, and the metric topology on (,) agrees with the topology on ... The metric space (,) is complete (every ...
In topology one considers completely metrizable spaces, spaces for which there exists at least one complete metric inducing the given topology. Completely metrizable spaces can be characterized as those spaces that can be written as an intersection of countably many open subsets of some complete metric space.
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.