Search results
Results from the WOW.Com Content Network
Functions to a metric space. If X is any set and M is a metric space, then the set of all bounded functions: (i.e. those functions whose image is a bounded subset of ) can be turned into a metric space by defining the distance between two bounded functions f and g to be (,) = ((), ()).
In general, when we define metric space the distance function is taken to be a real-valued function. The real numbers form an ordered field which is Archimedean and order complete. These metric spaces have some nice properties like: in a metric space compactness, sequential compactness and countable compactness are equivalent etc.
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (M, d) is a function f from M to itself, with the property that there is some real number < such that for all x and y in M,
An ultrametric space is a pair (M, d) consisting of a set M together with an ultrametric d on M, which is called the space's associated distance function (also called a metric). If d satisfies all of the conditions except possibly condition 4 then d is called an ultrapseudometric on M. An ultrapseudometric space is a pair (M, d) consisting of a ...
The function d is a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. Metrizable/Metrisable
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 ...
For any metric space M, it is possible to construct a complete metric space M′ (which is also denoted as ¯), which contains M as a dense subspace. It has the following universal property : if N is any complete metric space and f is any uniformly continuous function from M to N , then there exists a unique uniformly continuous function f ...
Hyperbolic metric space (3 P) M. Metric linear spaces (1 C) R. Riemannian manifolds (41 P) Pages in category "Metric spaces"