Explore: Metric Spaces

of a metric structure on the rational numbers. Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces. Many

distance metric. In contrast, infinite-dimensional normed vector spaces may or may not be complete; those that are complete are Banach spaces. The space C[a

examples of hyperbolic spaces are spaces with bounded diameter (for example finite or compact spaces) and the real line. Metric trees and more generally

metrics on Lie groups and homogeneous spaces are defined intrinsically by using group actions to transport an inner product on a single tangent space

theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise

separable Banach space, the Cantor space, and the Baire space. Additionally, some spaces that are not complete metric spaces in the usual metric may be Polish;

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance. These maps are the

metric spaces need not be the same) And since any Euclidean space is complete, we can thus conclude that all finite-dimensional normed vector spaces are

2003) Every separable metric space is isometric to a subset of the Urysohn universal space. For nonseparable spaces: A metric space of density equal to

space to be metrizable. Metrizable spaces inherit all topological properties from metric spaces. For example, they are Hausdorff paracompact spaces (and