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 - 5 of 14

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.
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.
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.
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.
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.

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