Applied General Topology - Vol 03, No 1 (2002)
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Tabla de contenidos
- Hausdorff compactifications and zero-one measures II
- A contribution to fuzzy subspaces
- Some results on best proximity pair theorems
- On paracompact spaces and projectively inductively closed functors
- All hypertopologies are hit-and-miss
- Minimal TUD spaces
- Every finite system of T1 uniformities comes from a single distance structure
- Fenestrations induced by perfect tilings
- Topological groups with dense compactly generated subgroups
- Duality and quasi-normability for complexity spaces
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- PublicationDuality and quasi-normability for complexity spaces(Universitat Politècnica de València, 2002-04-01) Romaguera, Salvador; Schellekens, M.P.; Ministerio de Ciencia y Tecnología[EN] The complexity (quasi-metric) space was introduced in [23] to study complexity analysis of programs. Recently, it was introduced in [22] the dual complexity (quasi-metric) space, as a subspace of the function space [0,) ω. Several quasi-metric properties of the complexity space were obtained via the analysis of its dual. We here show that the structure of a quasi-normed semilinear space provides a suitable setting to carry out an analysis of the dual complexity space. We show that if (E,) is a biBanach space (i.e., a quasi-normed space whose induced quasi-metric is bicomplete), then the function space (B*E, B* ) is biBanach, where B*E = {f : E Σ∞n=0 2-n( V ) } and B* = Σ∞n=0 2-n We deduce that the dual complexity space admits a structure of quasinormed semlinear space such that the induced quasi-metric space is order-convex, upper weightable and Smyth complete, not only in the case that this dual is a subspace of [0,)ω but also in the general case that it is a subspace of Fω where F is any biBanach normweightable space. We also prove that for a large class of dual complexity (sub)spaces, lower boundedness implies total boundedness. Finally, we investigate completeness of the quasi-metric of uniform convergence and of the Hausdorff quasi-pseudo-metric for the dual complexity space, in the context of function spaces and hyperspaces, respectively.
- PublicationTopological groups with dense compactly generated subgroups(Universitat Politècnica de València, 2002-04-01) Fujita, Hiroshi; Shakhmatov, Dimitri[EN] A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) 0-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both 0-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both 0-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.
- PublicationFenestrations induced by perfect tilings(Universitat Politècnica de València, 2002-04-01) Arenas, F.G.; Puertas, M.L.; Ministerio de Ciencia y Tecnología[EN] In this paper we study those regular fenestrations (as defined by Kronheimer in [3]) that are obtained from a tiling of a topological space. Under weak conditions we obtain that the canonical grid is also the minimal grid associated to each tiling and we prove that it is a T0-Alexandroff semirregular trace space. We also present some examples illustrating how the properties of the grid depend on the properties of the tiling and we pose some questions. Finally we study the topological properties of the grid depending on the properties of the space and the tiling.
- PublicationEvery finite system of T1 uniformities comes from a single distance structure(Universitat Politècnica de València, 2002-04-01) Heitzig, Jobst[EN] Using the general notion of distance function introduced in an earlier paper, a construction of the finest distance structure which induces a given quasi-uniformity is given. Moreover, when the usual defining condition xy : d(y; x) of the basic entourages is generalized to nd(y; x) n (for a fixed positive integer n), it turns out that if the value-monoid of the distance function is commutative, one gets a countably infinite family of quasi-uniformities on the underlying set. It is then shown that at least every finite system and every descending sequence of T1 quasi-uniformities which fulfil a weak symmetry condition is included in such a family. This is only possible since, in contrast to real metric spaces, the distance function need not be symmetric.
- PublicationMinimal TUD spaces(Universitat Politècnica de València, 2002-04-01) McCluskey, A.E.; Watson, Stephen[EN] A topological space is TUD if the derived set of each point is the union of disjoint closed sets. We show that there is a minimal TUD space which is not just the Alexandroff topology on a linear order. Indeed the structure of the underlying partial order of a minimal TUD space can be quite complex. This contrasts sharply with the known results on minimality for weak separation axioms.