Applied General Topology - Vol 05, No 2 (2004)
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- A fuzzification of the category of M-valued L-topological spaces
- On separation axioms of uniform bundles and sheaves
- On the cardinality of indifference classes
- Resolvability of ball structures
- Relative Collectionwise Normality
- Continuous representability of interval orders
- Which topologies can have immediate successors in the lattice of T1-topologies?
- Spaces whose Pseudocompact Subspaces are Closed Subsets
- Two transfinite chains of separation conditions between T1 and T2
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- PublicationA fuzzification of the category of M-valued L-topological spaces(Universitat Politècnica de València, 2004-10-01) Kubiak, Tomasz; Sostak, Alexander P.[EN] A fuzzy category is a certain superstructure over an ordinary category in which ”potential” objects and ”potential” morphisms could be such to a certain degree. The aim of this paper is to introduce a fuzzy category FTOP(L,M) extending the category TOP(L,M) of M-valued L- topological spaces which in its turn is an extension of the category TOP(L) of L-fuzzy topological spaces in Kubiak-Sostak’s sense. Basic properties of the fuzzy category FTOP(L,M) and its objects are studied.
- PublicationContinuous representability of interval orders(Universitat Politècnica de València, 2004-10-01) Candeal, Juan C.; Induráin, Esteban; Zudaire, M.; Ministerio de Educación y Cultura[EN] In the framework of the analysis of orderings whose associated indifference relation is not necessarily transitive, we study the structure of an interval order, and its representability through a pair of continuous real-valued functions. Inspired in recent characterizations of the representability of interval orders, we obtain new results concerning the existence of continuous real-valued representations. Classical results are also restated in a unified framework.
- PublicationOn separation axioms of uniform bundles and sheaves(Universitat Politècnica de València, 2004-10-01) Neira, Clara M.; Varela, Januario; Fundación Mazda para el Arte y la Ciencia, Colombia[EN] In the context of the theory of uniform bundles in the sense of J. Dauns and K. H. Hofmann, the topology of the fiber space of a uniform bundle depends on the assumption of upper semicontinuity of its defining set of pseudometrics when composed with local sections. In this paper we show that the additional hypothesis of lower semicontinuity of these functions secures that the fiber space of the uniform bundle is Hausdorff, regular or completely regular provided that the base space has the corresponding separation axiom. Similar results for the particular important case of sheaves of sets follow suit.
- PublicationOn the cardinality of indifference classes(Universitat Politècnica de València, 2004-10-01) Herden, Gerhard; Pallack, Andreas[EN] Let “≤” be a continuous total preorder on some topological space (X, t). Then the cardinality or at least lower and upper bounds of the cardinality of the indifference (equivalence) classes of “≤ ” will be computed. In addition, the relevance of these bounds in mathematical utility theory and the theory of orderable topological spaces will be discussed.
- PublicationRelative Collectionwise Normality(Universitat Politècnica de València, 2004-10-01) Grabner, Eliser; Grabner, Gary; Miyazaki, Kazumi; Tartir, Jamal[EN] In this paper we study properties of relative collectionwise normality type based on relative properties of normality type introduced by Arhangel’skii and Genedi. Theorem Suppose Y is strongly regular in the space X. If Y is paracompact in X then Y is collectionwise normal in X. Example A T2 space X having a subspace which is 1− paracompact in X but not collectionwise normal in X. Theorem Suppose that Y is s- regular in the space X. If Y is metacompact in X and strongly collectionwise normal in X then Y is paracompact in X.
- PublicationResolvability of ball structures(Universitat Politècnica de València, 2004-10-01) Protasov, Igor V.[EN] A ball structure is a triple B = (X, P, B) where X, P are nonempty sets and, for any x ∈ X, α ∈ P, B(x, α) is a subset of X, which is called a ball of radius α around x. It is supposed that x ∈ B(x, α) for any x ∈ X, α ∈ P. A subset Y C X is called large if X = B(Y, α) for some α ∈ P where B(Y, α) = Uy∈Y B(y, α). The set X is called a support of B, P is called a set of radiuses. Given a cardinal k, B is called k-resolvable if X can be partitioned to k large subsets. The cardinal res B = sup {k : B is k-resolvable} is called a resolvability of B. We determine the resolvability of the ball structures related to metric spaces, groups and filters.
- PublicationSpaces whose Pseudocompact Subspaces are Closed Subsets(Universitat Politècnica de València, 2004-10-01) Dow, Alan; Porter, Jack R.; Stephenson, R.M.; Grant Woods, R.; National Science Foundation, EEUU[EN] Every first countable pseudocompact Tychonoff space X has the property that every pseudocompact subspace of X is a closed subset of X (denoted herein by “FCC”). We study the property FCC and several closely related ones, and focus on the behavior of extension and other spaces which have one or more of these properties. Characterization, embedding and product theorems are obtained, and some examples are given which provide results such as the following. There exists a separable Moore space which has no regular, FCC extension space. There exists a compact Hausdorff Fréchet space which is not FCC. There exists a compact Hausdorff Fréchet space X such that X, but not X2, is FCC.
- PublicationTwo transfinite chains of separation conditions between T1 and T2(Universitat Politècnica de València, 2004-10-01) Sequeira, Luís[EN] Two new families of separation conditions have arisen in the study of the impact that the algebraic properties of topological algebras have on the topologies that may occur on their underlying spaces. We describe the relative strengths of these families of separation conditions for general spaces.
- PublicationWhich topologies can have immediate successors in the lattice of T1-topologies?(Universitat Politècnica de València, 2004-10-01) Alas, Ofelia T.; Wilson, Richard G.; Consejo Nacional de Ciencia y Tecnología, México[EN] We give a new characterization of those topologies which have an immediate successor or cover in the lattice of T1-topologies on a set and show that certain classes of compact and countably compact topologies do not have covers.