Completions and compactifications of quasi-uniform spaces

Romaguera, Salvador; Sánchez-Granero, M.A.

doi:10.1016/s0166-8641(01)00204-8

Cited by 10 publications

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“…The space is bicomplete and only its Doitchinov completion yields the expected completion. )The definition of Cauchy net or filter in quasi-uniform space which has been proposed in bicompleteness and D-completeness gives a completing theory which satisfies the above requirements given by Doitchinov.Császár in [8, page 228] and Deák in [2, page 412] have introduced the notion of half-completeness in a quasi-uniform space which generalizes the well-known notions of bicompleteness and D-completeness.Romaguera and Sánchez-Granero in [3] introduced the notions of -half-completion and -compactification of a T 1 quasi-uniform space (X,ᐁ). The authors use the notion of -half-completion to show that if a T 1 quasi-uniform space has a -compactification, then it is unique (up to quasi-uniformism).In this paper, we introduce the notions of weakly quiet and -weakly pair Cauchy bounded quasi-uniform spaces which generalize the notions of quiet and semisymmetric quasi-uniform spaces defined by Doitchinov [1] and Deák [2], respectively.…”

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“…Romaguera and Sánchez-Granero in [3] introduced the notions of -half-completion and -compactification of a T 1 quasi-uniform space (X,ᐁ). The authors use the notion of -half-completion to show that if a T 1 quasi-uniform space has a -compactification, then it is unique (up to quasi-uniformism).…”

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Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Andrikopoulos¹

2007

International Journal of Mathematics and Mathematical Sciences

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We generalize the notions of quietness and semisymmetry defined by Doitchinov (1991) and Deák (1991) and we study the role of these extended notions on the coincidence of some well-known quasi-uniform completeness. In particular, it is shown that the bicompletion coincides (up to quasi-uniformism) with the standard D-completion in quiet -weakly pair Cauchy bounded quasi-uniform spaces and it coincides with -half-completion defined by Romaguera and Sánchez-Granero (2002), in T 0 -weakly quiet quasiuniform spaces. definition of Cauchy net or Cauchy filter (often nets and filters lead to equivalent theories). The reason for the difficulty to develop a satisfactory theory of completeness for the class of all quasi-uniform spaces is due to the difficulty to approach the notion of Cauchy net (filter) in an appropriate way from the case of uniform spaces to the case of quasiuniform spaces. More precisely, since uniform spaces belong to the class of quasi-uniform spaces, according to Doitchinov [1], a notion of Cauchy net in any quasi-uniform space has to be defined in such a manner that this definition provides the properties that convergent nets are Cauchy, and it agrees with the usual definition for uniform spaces. Moreover, the suggested completion must be a monotone operator with respect to inclusion and give rise to the usual uniform completion in the uniform case.Fletcher and Lindgren in [4,5] give two equivalent theories of completeness, the pair completeness or bicompleteness. The first one uses pair of filters (Ᏺ,Ᏻ) for which Ᏺ ∪ Ᏻ is a filter base and the second is modelled upon the presentations of the completion of a uniform space given by Bourbaki [6]. It turns out that this concept coincides (for quasiuniform spaces) with that of double completeness introduced by Császár in [7]. The authors proved that the bicompletion has a very nice behavior: every T 0 quasi-uniform space can be quasi-uniformly embedded in a unique (i.e., up to quasi-uniformism) T 0 quasi-uniform space; for uniform spaces the construction coincides with the usual one. In [1], Doitchinov has introduced the notion of D-completeness and he proved that a certain category of quasi-uniform T 2 -spaces, the so-called quiet spaces, has a nice behavior: every quiet T 2 -space can be quasi-uniformly embedded in a unique D-complete quiet T 2 -space; for uniform spaces (which are always quiet) the construction coincides with the usual one. D-completeness is founded on a concept of a Cauchy net which can be done satisfying some most natural basic requirements. The Doitchinov completion provides another sophisticated completion that is in general larger than the bicompletion. (Consider for instance the set of the rationals equipped with the Sorgenfrey quasimetric. The space is bicomplete and only its Doitchinov completion yields the expected completion.)The definition of Cauchy net or filter in quasi-uniform space which has been proposed in bicompleteness and D-completeness gives a completing theory which satisfies the above requirements given by Doit...

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Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Andrikopoulos¹

2007

International Journal of Mathematics and Mathematical Sciences

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“…As it is mentioned in the paper [137], the authors did not succeed to obtain a satisfactory theory for these problems. Some progress in this direction was obtained by Alemany and Romaguera [8], Doichinov [43,47], Gregori and Romaguera [75], Romaguera and Sánchez-Granero [144]. We shall discuss in Section 3 the existence of a bicompletion for an asymmetric normed space, following the paper [63].…”

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Functional Analysis in Asymmetric Normed Spaces

Cobzaş

2013

Frontiers in Mathematics

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171

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The aim of this paper is to present a survey of some recent results obtained in the study of spaces with asymmetric norm. The presentation follows the ideas from the theory of normed spaces (topology, continuous linear operators, continuous linear functionals, duality, geometry of asymmetric normed spaces, compact operators) emphasizing similarities as well as differences with respect to the classical theory. The main difference comes form the fact that the dual of an asymmetric normed space X is not a linear space, but merely a convex cone in the space of all linear functionals on X. Due to this fact, a careful treatment of the duality problems (e.g. reflexivity) and of other results as, for instance, the extension of fundamental principles of functional analysis -the open mapping theorem and the closed graph theorem -to this setting, is needed. Contents 1. Introduction: 1.1 Quasi-metric spaces and asymmetric normed spaces; 1.2 The topology of a quasisemimetric space; 1.3 Quasi-uniform spaces.2. Completeness and compactness in quasi-metric and in quasi-uniform spaces: 2.1 Various notions of completeness for quasi-metric spaces; 2.2 Compactness, total bondedness and precompactness; 2.3 Completeness in quasi-uniform spaces; 2.4 Baire category.3. Continuous linear operators between asymmetric normed spaces: 3.1 The asymmetric norm of a continuous linear operator; 3.2 The normed cone of continuous linear operators -completeness; 3.3 The bicompletion of an asymmetric normed space; 3.4 Open mapping and closed graph theorems for asymmetric normed spaces; 3.5 Normed cones; 3.6 The w ♭ topology of the dual space X ♭ p ; 3.7 Compact subsets of an asymmetric normed space; 3.8 The conjugate operator, precompact operators between asymmetric normed spaces and a Schauder type theorem; 3.9 Asymmetric moduli of smoothness and rotundity; 3.10 Asymmetric topologies on normed lattices. 4. Linear functionals on an asymmetric normed space: 4.1 Some properties of continuous linear functionals; 4.2 Hahn-Banach type theorems; 4.3 The bidual space, reflexivity and Goldstine theorem. 5. The Minkowski functional and the separation of convex sets: 5.1 The Minkowski gauge functional -definition and properties; 5.2 The separation of convex sets; 5.3 Extremal points and Krein-Milman theorem.6. Applications to best approximation: 6.1 Characterizations of nearest points in convex sets and duality; 6.2 The distance to a hyperplane; 6.3 Best approximation by elements of sets with convex complement; 6.4 Optimal points; 6.5 Sign-sensitive approximation in spaces of continuous or integrable functions.7. Spaces of semi-Lipschitz functions: 7.1 Semi-Lipschitz functions -definition, the extension property, applications to best approximation in quasi-metric spaces; 7.2 Properties of the cone of semi-Lipschitz functions -linearity, completeness. ReferencesThis research was supported by Grant CNCSIS 2261, ID 543.Remark 1.1. Since the terms "quasi-norm", "quasi-normed space" and "quasi-Banach space" are already "registered trademarks" (see, for instance, th...

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“…Moreover, a point symmetric totally bounded T 1 quasi-uniform space may have many totally bounded compactifications (see [5, page 34]) . Contrary to this notion, Romaguera and Sánchez-Granero have introduced the notion of * -compactification of a T 1 quasiuniform space (see [8], [10] and [11]) and prove that: (a) Each T 1 quasi-uniform space having a T 1 * -compactification has an (up to quasi-isomorphism) unique T 1 * -compactification ([11, Corollary of Theorem 1]); and (b) All the Wallmantype compactifications of a T 1 topological space can be characterized in terms of the * -compactification of its point symmetric totally transitive compatible quasi-uniformities ([9, Theorem 1]). The proof of (a) is achieved with the help of the notion of T 1 * -half completion of a quasi-uniform space, which is introduced in [11].…”

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*-Compactifications of quasi-uniform spaces

Romaguera

Sánchez-Granero

2007

Studia Scientiarum Mathematicarum Hungarica

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Abstract.Romaguera and have introduced the notion of T1 * -half completion and used it to see when a quasi-uniform space has a * -compactification. In this paper, for any quasi-uniform space, we construct a * -half completion, called standard * -half completion. The constructed * -half completion coincides with the usual uniform completion in the uniform spaces and is the unique (up to quasi-isomorphism) T1* -half completion of a symmetrizable quasi-uniform space. Moreover, it constitutes a * -compactification for * -Cauchy bounded quasi-uniform spaces. Finally, we give an example which shows that the standard * -half completion differs from the bicompletion construction.2000 AMS Classification: 54E15, 54D35.

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Completions and compactifications of quasi-uniform spaces

Cited by 10 publications

References 9 publications

Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Symmetry Conditions on the Coincidence of Some Notions of Quasi-Uniform Completeness

Functional Analysis in Asymmetric Normed Spaces

*-Compactifications of quasi-uniform spaces

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