Applied General Topology - Vol 04, No 1 (2003)

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A note on separation and compactness in categories of convergence spaces
The quasitopos hull of the construct of closure spaces
On complete objects in the category of T0 closure spaces
On φ 1,2-countable compactness and filters
The local triangle axiom in topology and domain theory
Dense Sδ-diagonals and linearly ordered extensions
Holonomy, extendibility, and the star universal cover of a topological groupoid
Functorial approach structures
On the use of partial orders in uniform spaces
Effective representations of the space of linear bounded operators
Skew compact semigroups
On classes of T0 spaces admitting completions
Di-uniform texture spaces
Injective locales over perfect embeddings and algebras of the upper powerlocale monad

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Now showing 1 - 14 of 14

Holonomy, extendibility, and the star universal cover of a topological groupoid
(Universitat Politècnica de València, 2003-04-01) Mucuk, Osman; Icen, Ilhan
[EN] Let G be a groupoid and W be a subset of G which contains all the identities and has a topology. With some conditions on G and W, the pair (G;W) is called a locally topological groupoid. We explain a criterion for a locally topological groupoid to be extendible to a topological groupoid. In this paper we apply this result to get a topology on the monodromy groupoid MG which is the union of the universal covers of Gx's.
The local triangle axiom in topology and domain theory
(Universitat Politècnica de València, 2003-04-01) Waszkiewicz, Pawel
[EN] We introduce a general notion of distance in weakly separated topological spaces. Our approach differs from existing ones since we do not assume the reflexivity axiom in general. We demonstrate that our partial semimetric spaces provide a common generalization of semimetrics known from Topology and both partial metrics and measurements studied in Quantitative Domain Theory. In the paper, we focus on the local triangle axiom, which is a substitute for the triangle inequality in our distance spaces. We use it to prove a counterpart of the famous Archangelskij Metrization Theorem in the more general context of partial semimetric spaces. Finally, we consider the framework of algebraic domains and employ Lebesgue measurements to obtain a complete characterization of partial metrizability of the Scott topology.
A note on separation and compactness in categories of convergence spaces
(Universitat Politècnica de València, 2003-04-01) Baran, Mehmet; Kula, Muammer
[EN] In previous papers, various notions of compact, T3, T4, and Tychonoff objects in a topological category were introduced and compared. The main objective of this paper is to characterize each of these classes of objects in the categories of filter and local filter convergence spaces as well as to examine how these various generalizations are related.
On φ 1,2-countable compactness and filters
(Universitat Politècnica de València, 2003-04-01) Yalvaç, T.H.
[EN] In this work the author investigates some relations between φ 1,2-countable compactness, filters, sequences and φ 1,2-closure operators.
The quasitopos hull of the construct of closure spaces
(Universitat Politècnica de València, 2003-04-01) Claes, Veerle; Sonck, G.
[EN] In the list of convenience properties for topological constructs the property of being a quasitopos is one of the most interesting ones for investigations in function spaces, differential calculus, functional analysis, homotopy theory, etc. The topological construct Cls of closure spaces and continuous maps is not a quasitopos. In this article we give an explicit description of the quasitopos topological hull of Cls using a method of F. Schwarz: we first describe the extensional topological hull of Cls and of this hull we construct the cartesian closed topological hull.
Functorial approach structures
(Universitat Politècnica de València, 2003-04-01) Brümmer, Guillaume C.L.; Sioen, M.
[EN] We show that there exists at least a proper class of functorial approach structures, i.e., right inverses to the forgetful functor T : AP→ Top (where AP denotes the topological construct of approach spaces and contractions as introduced by R. Lowen). There is however a great difference in nature of these functorial approach structures when compared to the quasi-uniform paradigm which has been extensively studied by the first author: whereas it is well-known from [2] that a large class of epireflective subcategories of Top0 can be “parametrized” using the interaction of functorial quasi-uniformities with the quasi-uniform bicompletion, we show that using functorial approach structures together with the approach bicompletion developed in [10], only Top0 itself can be retrieved in this way.
Effective representations of the space of linear bounded operators
(Universitat Politècnica de València, 2003-04-01) Brattka, Vasco; Deutsche Forschungsgemeinschaft
[EN] Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition. We discuss a number of representations of the space of linear bounded operators on a Banach space. Since the operator norm topology of the operator space is nonseparable in typical cases, the operator space cannot be represented admissibly with respect to this topology. However, other topologies, like the compact open topology and the Fell topology (on the operator graph) give rise to a number of promising representations of operator spaces which can partially replace the operator norm topology. These representations reflect the information which is included in certain data structures for operators, such as programs or enumerations of graphs. We investigate the sublattice of these representations with respect to continuous and computable reducibility. Certain additional conditions, such as finite dimensionality, let some classes of representations collapse, and thus, change the corresponding graph. Altogether, a precise picture of possible data structures for operator spaces and their mutual relation can be drawn.
Skew compact semigroups
(Universitat Politècnica de València, 2003-04-01) Kopperman, Ralph; Robbie, Desmond
[EN] Skew compact spaces are the best behaving generalization of compact Hausdorff spaces to non-Hausdorff spaces. They are those (X ; τ ) such that there is another topology τ* on X for which τ V τ* is compact and (X; τ ; τ*) is pairwise Hausdorff; under these conditions, τ uniquely determines τ *, and (X; τ*) is also skew compact. Much of the theory of compact T2 semigroups extends to this wider class. We show: A continuous skew compact semigroup is a semigroup with skew compact topology τ, such that the semigroup operation is continuous τ2→ τ. Each of these contains a unique minimal ideal which is an upper set with respect to the specialization order. A skew compact semigroup which is a continuous semigroup with respect to both topologies is called a de Groot semigroup. Given one of these, we show: It is a compact Hausdorff group if either the operation is cancellative, or there is a unique idempotent and S2 = S. Its topology arises from its subinvariant quasimetrics. Each *-closed ideal ≠ S is contained in a proper open ideal.
On classes of T0 spaces admitting completions
(Universitat Politècnica de València, 2003-04-01) Giuli, Eraldo
[EN] For a given class X of T0 spaces the existence of a subclass C, having the same properties that the class of complete metric spaces has in the class of all metric spaces and non-expansive maps, is investigated. A positive example is the class of all T0 spaces, with C the class of sober T0 spaces, and a negative example is the class of Tychonoff spaces. We prove that X has the previous property (i.e., admits completions) whenever it is the class of T0 spaces of an hereditary coreflective subcategory of a suitable supercategory of the category Top of topological spaces. Two classes of examples are provided.
Di-uniform texture spaces
(Universitat Politècnica de València, 2003-04-01) Ozcag, Selma; Brown, Lawrence M.
[EN] Textures were introduced by the second author as a point-based setting for the study of fuzzy sets, and have since proved to be an appropriate framework for the development of complement-free mathematical concepts. In this paper the authors lay the foundation for a theory of uniformities in a textural context. Analogues are given for both the diagonal and covering approaches to the classical theory of uniform structures, the notion of uniform topology is generalized and an analogue given for the well known result that a topological space is uniformizable if and only if it is completely regular. Finally a textural analogue of the classical interplay between uniformities and families of pseudo-metrics is presented.
Injective locales over perfect embeddings and algebras of the upper powerlocale monad
(Universitat Politècnica de València, 2003-04-01) Escardó, Martín
[EN] We show that the locales which are injective over perfect sublocale embeddings coincide with the underlying objects of the algebras of the upper powerlocale monad, and we characterize them as those whose frames of opens enjoy a property analogous to stable supercontinuity.
Dense Sδ-diagonals and linearly ordered extensions
(Universitat Politècnica de València, 2003-04-01) Hosobuchi, Masami
[EN] The notion of the Sδ-diagonal was introduced by H. R. Bennett to study the quasi-developability of linearly ordered spaces. In an earlier paper, we obtained a characterization of topological spaces with an Sδ-diagonal and we showed that the Sδ-diagonal property is stronger than the quasi-Gδ-diagonal -diagonal property. In this paper, we define a dense Sδ-diagonal of a space and show that two linearly ordered extensions of a generalized ordered space X have dense Sδ-diagonals if the sets of right and left looking points are countable.
On complete objects in the category of T0 closure spaces
(Universitat Politècnica de València, 2003-04-01) Deses, D.; Giuli, Eraldo; Lowen-Colebunders, E.
[EN] In this paper we present an example in the setting of closure spaces that fits in the general theory on “complete objects” as developed by G. C. L. Brümmer and E. Giuli. For V the class of epimorphic embeddings in the construct Cl0 of T0 closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the Vinjective objects as “complete” ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins.
On the use of partial orders in uniform spaces
(Universitat Politècnica de València, 2003-04-01) Burdick, Bruce S.
[EN] We investigate the use of nets indexed by preorders in uniform spaces. Nine different Cauchy conditions and four different convergence conditions yield 36 completeness properties, each of which turns out to be equivalent to a known form of completeness. We also use these preordered nets to characterize the functors θ, λ, and v, which are associated with these completeness properties. In the case of λ we give an example to show that the analogous characterization with predirected nets does not work.

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