Uniform space | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-27

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Uniform_space


00:01:06 1 Definition
00:01:23 1.1 Entourage definition
00:09:21 1.2 Pseudometrics definition
00:11:10 1.3 Uniform cover definition
00:12:27 2 Topology of uniform spaces
00:13:38 2.1 Uniformizable spaces
00:16:09 3 Uniform continuity
00:17:00 4 Completeness
00:18:49 4.1 Hausdorff completion of a uniform space
00:19:24 5 Examples
00:20:57 6 History
00:21:04 7 See also
00:22:42 8 References
00:24:13 History
00:24:51 See also



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SUMMARY
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In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence.
The conceptual difference between uniform and topological structures is that, in a uniform space, one can formalize certain notions of relative closeness and closeness of points.  In other words, ideas like "x is closer to a than y is to b" make sense in uniform spaces.  By comparison, in a general topological space, given sets A,B it is meaningful to say that a point x is arbitrarily close to A (i.e., in the closure of A), or perhaps that A is a smaller neighborhood of x than B, but notions of closeness of points and relative closeness are not described well by topological structure alone.
Uniform spaces generalize metric spaces and topological groups and therefore underlie most of analysis.