Bounded metric space | Wikipedia audio article

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00:01:37 1 History
00:01:59 2 Definition
00:04:37 3 Examples of metric spaces
00:19:26 4 Open and closed sets, topology and convergence
00:22:49 5 Types of metric spaces
00:23:00 5.1 Complete spaces
00:25:53 5.2 Bounded and totally bounded spaces
00:27:44 5.3 Compact spaces
00:29:04 5.4 Locally compact and proper spaces
00:29:42 5.5 Connectedness
00:31:37 5.6 Separable spaces
00:32:08 5.7 Pointed metric spaces
00:33:33 6 Types of maps between metric spaces
00:33:52 6.1 Continuous maps
00:35:15 6.2 Uniformly continuous maps
00:36:11 6.3 Lipschitz-continuous maps and contractions
00:37:00 6.4 Isometries
00:38:16 6.5 Quasi-isometries
00:40:12 7 Notions of metric space equivalence
00:41:12 8 Topological properties
00:42:21 9 Distance between points and sets; Hausdorff distance and Gromov metric
00:46:29 10 Product metric spaces
00:50:56 10.1 Continuity of distance
00:52:07 11 Quotient metric spaces
00:57:56 12 Generalizations of metric spaces
00:59:54 12.1 Metric spaces as enriched categories
01:04:23 13 See also



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SUMMARY
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In mathematics, a metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally:

the distance from a point to itself is zero,
the distance between two distinct points is positive,
the distance from A to B is the same as the distance from B to A, and
the distance from A to B (directly) is less than or equal to the distance from A to B via any third point C.A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.
The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities.