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| CATEGORIES ABOUT UNIFORM SPACE | |
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The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "''x'' is as close to ''a'' as ''y'' is to ''b''", while in a topological space you can only formalize "''x'' is as close to ''a'' as ''y'' is to ''a''". Uniform spaces generalize Metric Space s and Topological Group s and therefore underlie most of Analysis . HISTORY Before André Weil gave the first explicit definition of a uniform structure in 1937 , uniform concepts, like completeness, were discussed using Metric Spaces . Nicolas Bourbaki provided the definition of uniform structure in terms of entourages in their book Topologie Générale and John Tukey gave the uniform cover definition. Weil also characterized uniform spaces in terms of a family of pseudometrics. DEFINITION Entourage definition A uniform space (''X'',Φ) is a Set ''X'' equipped with a nonempty family of Subsets of the Cartesian Product ''X'' × ''X'' (Φ is called the '''uniform structure''' of ''X'' and its elements '''entourages''' ( French :neighborhoods or ''surroundings'') with the following properties # if ''U'' is in Φ, then ''U'' contains the diagonal { (''x'', ''x'') : ''x'' in ''X'' }. # if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ # if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ # if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''. # if ''U'' is in Φ, then { (''y'', ''x'') : (''x'', ''y'') in ''U'' } is also in Φ If the last property is omitted we call the space quasiuniform. One usually writes ''U'' : (''x'',''y'')∈''U''}. On a graph, a typical entourage is drawn as a blob surrounding the "''y''=''x''" diagonal. The ''U''[''x'' ’s are the vertical cross-sections. ''U'' will be a typical neighbourhood of ''x''. ''U''[''y'' will then be a typical neighborhood of ''y''. Unlike a topological space, one can go further and treat ''U'' and ''U''[''y'' as having the same size ''U''. Uniform cover definition
# {X} is a uniform cover.
# If P and '''Q''' are uniform covers, then there is a uniform cover '''R''' that star-refines both P and '''Q'''. Given a point ''x'' and a uniform cover P, one can consider the union of the members of P that contain ''x'' as a typical neighbourhood of ''x'' of size "P", and this intuitive measure applies uniformly over the space. Given a uniform space in the entourage sense, define a cover P to be uniform if there is some entourage ''U'' such that for each ''x''∈''X'', there is an ''A''∈P such that ''U'' {Link without Title} ⊆''A''. These uniform covers form a uniform space as in the second definition. Conversely, given a uniform space in the uniform cover sense, the supersets of ∪{''A''×''A'' : ''A''∈P}, as P ranges over the uniform covers, are the entourages for a uniform space as in the first definition. Moreover, these two transformations are inverses of each other. Pseudometrics definition Uniform spaces may be defined alternatively and equivalently using systems of Pseudometrics , an approach which is often useful in Functional Analysis . INTUITION | ||
| Similarly, Metric Intuitions Transfer To Uniformity By Thinking Of ''a''&isin''U'' | "''x''" class="copylinks" target="_blank">{Link without Title} as a substitute for ''x''&minus''a''<&delta The &delta-&epsilon definition of uniform continuity translates directly into the uniform space definition The difference is that the topological sense of closeness given by ''O'' applies near ''x'' only, while the uniform sense of closeness given by ''U'' applies to the whole space |
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| Using Metrics, A Simple Example Of Distinct Uniform Structures With Coinciding Topologies Can Be Constructed For Instance, Let ''d''<sub>1</sub>(''x'',''y'') | ''x &minus y'' be the usual metric on '''R''' and let ''d''<sub>2</sub>(''x'',''y'') = ''e<sup>x</sup> &minus e<sup>y</sup>'' Then both metrics induce the usual topology on '''R''', yet the uniform structures are distinct, since { (x,y) : x &minus y < 1 } is an entourage in the uniform structure for ''d''<sub>1</sub> but not for ''d''<sub>2</sub> Informally, this example can be seen as taking the usual uniformity and distorting it through the action of a continuous yet non-uniformly continuous function |
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