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Document Type |
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Article In Journal |
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Document Title |
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on a certain uniform structure على هيكل موحد معينة |
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Subject |
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on a certain uniform structure |
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Document Language |
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English |
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Abstract |
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As with metric spaces, every uniform space X has a Hausdorff completion: that is, there exists a complete Hausdorff uniform space Y and a uniformly continuous map i: X → Y with the following property:
for any uniformly continuous mapping f of X into a complete Hausdorff uniform space Z, there is a unique uniformly continuous map g: Y → Z such that f = gi.
The Hausdorff completion Y is unique up to isomorphism. As a set, Y can be taken to consist of the minimal Cauchy filters on X. As the neighbourhood filter B(x) of each point x in X is a minimal Cauchy filter, the map i can be defined by mapping x to B(x). The map i thus defined is in general not injective; in fact, the graph of the equivalence relation i(x) = i(x ') is the intersection of all entourages of X, and thus i is injective precisely when X is Hausdorff.
The uniform structure on Y is defined as follows: for each symmetric entourage V (i.e., such that (x,y) is in V precisely when (y,x) is in V), let C(V) be the set of all pairs (F,G) of minimal Cauchy filters which have in common at least one V-small set. The sets C(V) can be shown to form a fundamental system of entourages; Y is equipped with the uniform structure thus defined.
The set i(X) is then a dense subset of Y. If X is Hausdorff, then i is an isomorphism onto i(X), and thus X can be identified with a dense subset of its completion. Moreover, i(X) is always Hausdorff; it is called the Hausdorff uniform space associated with X. If R denotes the equivalence relation i(x) = i(x '), then the quotient space X/R is homeomorphic to i(X). |
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ISSN |
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1012-5965 |
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Journal Name |
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Delta journal of science |
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Volume |
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29 |
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Issue Number |
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1 |
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Publishing Year |
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1425 AH
2005 AD |
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Article Type |
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Article |
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Added Date |
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Wednesday, March 20, 2013 |
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Researchers
| سهام جلال الصياد | sayad, Siham Jalal | Investigator | Doctorate | |
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