Applied General Topology - Vol 04, No 2 (2003)
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- Multivalued function spaces and Atsuji spaces
- Unusual and bijectively related manifolds
- Transitivity of hereditarily metacompact spaces
- Extensions of closure spaces
- Quasi-pseudometric properties of the Nikodym-Saks space
- Closure properties of function spaces
- Locally convex approach spaces
- A short note on hit-and-miss hyperspaces
- A better framework for first countable spaces
- Bounded point evaluations for cyclic Hilbert space operators
- Five different proofs of extraresolvability of countable totally bounded groups
- Groups with a small set of generators
- Developable hyperspaces are metrizable
- Orderability and continuous selections for Wijsman and Vietoris hyperspaces
- Paths in hyperspaces
- Hyperconvergences
- Bombay hypertopologies
- Graph topologies on closed multifunctions
- n-Tuple relations and topologies on function spaces
- A note on separation in AP
- Some properties of the containing spaces and saturated classes of spaces
- Density topology and pointwise convergence
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- PublicationOrderability and continuous selections for Wijsman and Vietoris hyperspaces(Universitat Politècnica de València, 2003-10-01) Di Caprio, Debora; Watson, Stephen[EN] Bertacchi and Costantini obtained some conditions equivalent to the existence of continuous selections for the Wijsman hyperspace of ultrametric Polish spaces. We introduce a new class of hypertopologies, the macro-topologies. Both the Wijsman topology and the Vietoris topology belong to this class. We show that subject to natural conditions, the base space admits a closed order such that the minimum map is a continuous selection for every macro-topology. In the setting of Polish spaces, these conditions are substantially weaker than the ones given by Bertacchi and Costantini. In particular, we conclude that Polish spaces satisfying these conditions can be endowed with a compatible order and that the minimum function is a continuous selection for the Wijsman topology, just as it is for [0; 1]. This also solves a problem implicitely raised in Bertacchi and Costantini's paper.
- PublicationBounded point evaluations for cyclic Hilbert space operators(Universitat Politècnica de València, 2003-10-01) Bourhim, A.; Abdus Salam International Centre for Theoretical Physics[EN] In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.
- PublicationFive different proofs of extraresolvability of countable totally bounded groups(Universitat Politècnica de València, 2003-10-01) Artico, Giuliano; Malykhin, Viatcheslav I.; Marconi, Umberto; Università degli studi di Padova[EN] We give different proofs of extraresolvability for countably in finite topological spaces and in particular for totally bounded groups.
- PublicationExtensions of closure spaces(Universitat Politècnica de València, 2003-10-01) Deses, D.; de Groot-Van der Voorde, A.; Lowen-Colebunders, E.[EN] A closure space X is a set endowed with a closure operator P(X) → P(X), satisfying the usual topological axioms, except finite additivity. A T1 closure extension Y of a closure space X induces a structure ϒ on X satisfying the smallness axioms introduced by H. Herrlich [?], except the one on finite unions of collections. We'll use the word seminearness for a smallness structure of this type, i.e. satisfying the conditions (S1),(S2),(S3) and (S5) from [?]. In this paper we show that every T1 seminearness structure ϒ on X can in fact be induced by a T1 closure extension. This result is quite different from its topological counterpart which was treated by S.A. Naimpally and J.H.M. Whitfield in [?]. Also in the topological setting the existence of (strict) extensions satisfying higher separation conditions such as T2 and T3 has been completely characterized by means of concreteness, separatedness and regularity [?]. In the closure setting these conditions will appear to be too weak to ensure the existence of suitable (strict) extensions. In this paper we introduce stronger alternatives in order to present internal characterizations of the existence of (strict) T2 or strict regular closure extensions.
- PublicationHyperconvergences(Universitat Politècnica de València, 2003-10-01) Dolecki, Szymon; Mynard, Frédéric[EN] The hyperconvergence (upper Kuratowski convergence) is the coarsest convergence on the set of closed subsets of a convergence space that makes the canonical evaluation continuous. Sundry reective and coreective properties of hyperconvergences are characterized in terms of the underlying convergence.
- PublicationGraph topologies on closed multifunctions(Universitat Politècnica de València, 2003-10-01) Di Maio, Giuseppe; Meccariello, Enrico; Naimpally, Somashekhar[EN] In this paper we study function space topologies on closed multifunctions, i.e. closed relations on X x Y using various hypertopologies. The hypertopologies are in essence, graph topologies i.e topologies on functions considered as graphs which are subsets of X x Y . We also study several topologies, including one that is derived from the Attouch-Wets filter on the range. We state embedding theorems which enable us to generalize and prove some recent results in the literature with the use of known results in the hyperspace of the range space and in the function space topologies of ordinary functions.
- PublicationTransitivity of hereditarily metacompact spaces(Universitat Politècnica de València, 2003-10-01) Künzi, Hans-Peter A.; Swiss National Science Foundation[EN] We prove that each regular hereditarily metacompact (monotonic) β-space has the property that the third power of any neighbornet belongs to its point-finite quasi-uniformity.
- PublicationUnusual and bijectively related manifolds(Universitat Politècnica de València, 2003-10-01) Hocking, John G.[EN] A manifold is “unusual” if it admits of a continuous self-bijection which is not a homeomorphism. The present paper is a survey of work published over yearsaugmented with recent examples and results
- PublicationSome properties of the containing spaces and saturated classes of spaces(Universitat Politècnica de València, 2003-10-01) Iliadis, Stavros[EN] Subjects of this paper are: (a) containing spaces constructed in [2] for an indexed collection S of subsets, (b) classes consisting of ordered pairs (Q,X), where Q is a subset of a space X, which are called classes of subsets, and (c) the notion of universality in such classes. We show that if T is a containing space constructed for an indexed collection S of spaces and for every X ϵ S, QX is a subset of X, then the corresponding containing space TIQ constructed for the indexed collection Q ={QX : X ϵ S} of spaces, under a simple condition, can be considered as a specific subset of T. We prove some “commutative” properties of these specific subsets. For classes of subsets we introduce the notion of a (properly) universal element and define the notion of a (complete) saturated class of subsets. Such a class is “saturated” by (properly) universal elements. We prove that the intersection of (complete) saturated classes of subsets is also a (complete) saturated class. We consider the following classes of subsets: (a) IP(Cl), (b) IP(Op), and (c) IP(n.dense) consisting of all pairs (Q;X) such that: (a) Q is a closed subset of X, (b) Q is an open subset of X, and (c) Q is a never dense subset of X, respectively. We prove that the classes IP(Cl) and IP(Op) are complete saturated and the class IP(n.dense) is saturated. Saturated classes of subsets are convenient to use for the construction of new saturated classes by the given ones.
- PublicationGroups with a small set of generators(Universitat Politècnica de València, 2003-10-01) Dikranjan, Dikran; Marconi, Umberto; Moresco, Roberto; Università degli studi di Padova; Ministero dell'università e della ricerca scientifica e tecnologica, Italia[EN] Following [22] we study the class S of all groups that admit a small set of generators. Here we adopt also another notion of smallness (P-small) introduced by Prodanov in the case of abelian groups. We push further some results obtained in [22] (by adding some new members of S) and partially resolve an open question posed in [22]. We show that in most cases the groups in S admit a P-small set of generators.
- PublicationA better framework for first countable spaces(Universitat Politècnica de València, 2003-10-01) Preuss, Gerhard[EN] In the realm of semiuniform convergence spaces first countability is divisible and leads to a well-behaved topological construct with natural function spaces and one-point extensions such that countable products of quotients are quotients. Every semiuniform convergence space (e.g. symmetric topological space, uniform space, filter space, etc.) has an underlying first countable space. Several applications of first countability in a broader context than the usual one of topological spaces are studied.
- PublicationDevelopable hyperspaces are metrizable(Universitat Politècnica de València, 2003-10-01) Holá, L'Ubica; Pelant, Jan; Zsilinszky, László; Czech Science Foundation[EN] Developability of hyperspace topologies (locally finite, (bounded) Vietoris, Fell, respectively) on the nonempty closed sets is characterized. Submetrizability and having a Gδ-diagonal in the hyperspace setting is also discussed.
- PublicationMultivalued function spaces and Atsuji spaces(Universitat Politècnica de València, 2003-10-01) Naimpally, Somashekhar[EN] In this paper we present two themes. The first one describes a transparent treatment of some of the recent results in graph topologies on multi-valued functions. The study includes Vietoris topology, Fell topology, Fell uniform topology on compacta and uniform topology on compacta. The second theme concerns when continuity is equivalent to proximal continuity or uniform continuity
- PublicationClosure properties of function spaces(Universitat Politècnica de València, 2003-10-01) Kocinac, Ljubisa D.R.; Ministry of Science, Technology and Development, Serbia[EN] In this paper we investigate some closure properties of the space Ck(X) of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology.
- Publicationn-Tuple relations and topologies on function spaces(Universitat Politècnica de València, 2003-10-01) Georgiou, D.N.; Iliadis, S.D.; Papadopoulos, B.K.[EN] In some results concerning S-splitting, S-jointly continuous, D-splitting and D-jointly continuous topologies are considered, where S and D are the Sierpinski space and the double-point space, respectively. Here we generalize these results replacing the spaces S and D by any finite space.
- PublicationA short note on hit-and-miss hyperspaces(Universitat Politècnica de València, 2003-10-01) Bartsch, René; Poppe, Harry[EN] Based on some set-theoretical observations, compactness results are given for general hit-and-miss hyperspaces. Compactness here is sometimes viewed splitting into “κ-Lindelöfness” and “κ-compactness” for cardinals κ. To focus only hit-and-miss structures, could look quite old-fashioned, but some importance, at least for the techniques, is given by a recent result, [8], of Som Naimpally, to who this article is hearty dedicated.
- PublicationQuasi-pseudometric properties of the Nikodym-Saks space(Universitat Politècnica de València, 2003-10-01) Ferrer, Jesús; Ministerio de Ciencia y Tecnología; European Regional Development Fund[EN] For a non-negative finite countably additive measure μ defined on the σ-field Σ of subsets of Ω, it is well known that a certain quotient of Σ can be turned into a complete metric space Σ (Ω), known as the Nikodym-Saks space, which yields such important results in Measure Theory and Functional Analysis as Vitali-Hahn-Saks and Nikodym's theorems. Here we study some topological properties of Σ (Ω) regarded as a quasi-pseudometric space.
- PublicationPaths in hyperspaces(Universitat Politècnica de València, 2003-10-01) Constantini, Camillo; Kubís, Wieslaw; State Committee for Scientific Research, Polonia[EN] We prove that the hyperspace of closed bounded sets with the Hausdor_ topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.
- PublicationLocally convex approach spaces(Universitat Politècnica de València, 2003-10-01) Sioen, M.; Verwulgen, S.[EN] We continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. We prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. Furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces.
- PublicationBombay hypertopologies(Universitat Politècnica de València, 2003-10-01) Di Maio, Giuseppe; Meccariello, Enrico; Naimpally, Somashekhar[EN] Recently it was shown that, in a metric space, the upper Wijsman convergence can be topologized with the introduction of a new far-miss topology. The resulting Wijsman topology is a mixture of the ball topology and the proximal ball topology. It leads easily to the generalized or g-Wijsman topology on the hyperspace of any topological space with a compatible LO-proximity and a cobase (i.e. a family of closed subsets which is closed under finite unions and which contains all singletons). Further generalization involving a topological space with two compatible LO-proximities and a cobase results in a new hypertopology which we call the Bombay topology. The generalized locally finite Bombay topology includes the known hypertopologies as special cases and moreover it gives birth to many new hypertopologies. We show how it facilitates comparison of any two hypertopologies by proving one simple result of which most of the existing results are easy consequences.