Applied General Topology - Vol 11, No 2 (2010)
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Tabla de contenidos
- Convergence semigroup categories
- Thin subsets of balleans
- Closed injective systems and its fundamental limit spaces
- The Alexandroff property and the preservation of strong uniform continuity
- Random selection of Borel sets
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- PublicationRandom selection of Borel sets(Universitat Politècnica de València, 2010-10-01) Günther, Bernd[EN] A theory of random Borel sets is presented, based on dyadic resolutions of compact metric spaces. The conditional expectation of the intersection of two independent random Borel sets is investigated. An example based on an embedding of Sierpinski’s universal curve into the space of Borel sets is given.
- PublicationThe Alexandroff property and the preservation of strong uniform continuity(Universitat Politècnica de València, 2010-10-01) Beer, Gerald[EN] In this paper we extend the theory of strong uniform continuity and strong uniform convergence, developed in the setting of metric spaces in, to the uniform space setting, where again the notion of shields plays a key role. Further, we display appropriate bornological/variational modifications of classical properties of Alexandroff [1] and of Bartle for nets of continuous functions, that combined with pointwise convergence, yield continuity of the limit for functions between metric spaces.
- PublicationClosed injective systems and its fundamental limit spaces(Universitat Politècnica de València, 2010-10-01) Fenille, Marcio Colombo[EN] In this article we introduce the concept of limit space and fundamental limit space for the so-called closed injected systems of topological spaces. We present the main results on existence anduniqueness of limit spaces and several concrete examples. In the main section of the text, we show that the closed injective system can be considered as objects of a category whose morphisms are the so-called cismorphisms. Moreover, the transition to fundamental limit space can be considered a functor from this category into the category of topological spaces and continuous maps. Later, we show results about properties on functors and counter-functors for inductive closed injective systemand fundamental limit spaces. We finish with the presentation of some results of characterization of fundamental limit spaces for some special systems and the study of the so-called perfect properties.
- PublicationThin subsets of balleans(Universitat Politècnica de València, 2010-10-01) Lutsenko, Ievgen; Protasov, Igor V.[EN] A ballean is a set endowed with some family of balls in such a way that a ballean can be considered as an asymptotic counterpar tof a uniform topological space. We characterize the ideal generated by the family of all thin subsets in an ordinal ballean, and apply this characterization to metric spaces and groups.
- PublicationConvergence semigroup categories(Universitat Politècnica de València, 2010-10-01) Richardson, Gary; Mikusinski, Piotr; Boustique, H.[EN] Properties of the category consisting of all objects of the form (X, S, λ) are investigated, where X is a convergence space, S is a commutative semigroup, and λ: X × S → X is a continuous action. A “generalized quotient” of each object is defined without making the usual assumption that for each fixed g ∈ S, λ(., g) : X → X is an injection.