Applied General Topology - Vol 02, No 2 (2001)
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- Coreflectively modified continuous duality applied to classical product theorems
- Topological normal forms of high degree for the simplest bifurcations
- The chainable continua are the spaces approximated by finite COTS
- Generalized closed sets: a unified approach
- Directed GF-spaces
- Selections and order-like relations
- On strongly reflexive topological groups
- Unions of chains of subgroups of a topologucal group
- Continuous maps in the Bohr Topology
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- PublicationContinuous maps in the Bohr Topology(Universitat Politècnica de València, 2001-10-01) Dikranjan, Dikran; Ministero dell'università e della ricerca scientifica e tecnologica, Italia[EN] The Bohr topology of an Abelian group G is the initial topology on G with respect to the family of all homomorphisms of G into the circle group. The group G equipped with the Bohr topology is denoted by G#. It was an open question of van Douwen whether for any two discrete abelian groups G and H of the same cardinality the topological spaces G# and H# are homeomorphic. A negative solution to van Douwen's problem was given independently by Kunen and by Watson and the authot. In both cases infinite dimensional vector spaces Vp over the finite field Zp were used to show that there is no homeomorphism between V#p and V#q for p not = q and |Vp| = |Vq|. More precisely, it was shown that every continuous map V#p -> V#q is more constant on an infinite subset of Vp hence cannot be a homeomorphism. Motivated by this phenomenon we establish in this paper the "typical" behavior of a continuous mpa f:V#2 -> H# (and discuss without proofs the more gnereal case f:V#p ->H#). The specific choice of p=2 permits to consider V2 as the set of all finite subsets of an infinite set B (the base of V2). A special attention wil be paid to the restriction of f to the doubletons and the four element subsets of B.
- PublicationUnions of chains of subgroups of a topologucal group(Universitat Politècnica de València, 2001-10-01) Torres Falcón, Yolanda[EN] We consider the following problem: If a topological group G is the union of an increasing chain of subgroups and certain cardinal invariants of the subgroups in the chain are known, what can be said about G? We prove that if the index of boundedness of each subgroup is strictly less than λ for some infinite cardinal λ, then the index of boundedness of G is at most λ. We also prove that if both the index of boundedness and the pseudocharacter of each subgroup in the chain are at most λ and G is countably compact, then │G│≤2 λ. Finally, we show that the last assertion is not valid in general, not even for pseudocompact groups.
- PublicationOn strongly reflexive topological groups(Universitat Politècnica de València, 2001-10-01) Chasco, M. J.; Martin-Peinador, E.; Dirección General de Investigación Científica y Técnica[EN] An Abelian topological group G is strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of its dual group G⋀, is reflexive. In this paper we prove the following: the annihilator of a closed subgroup of an almost metrizable group is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of the starting group. As a consequence of this perfect duality, an almost metrizable group is strongly reflexive just if its Hausdorff quotients, as well as the Hausdorff quotients of its dual, are reflexive. The simplification obtained may be significant from an operative point of view.
- PublicationSelections and order-like relations(Universitat Politècnica de València, 2001-10-01) Gutev, Valentin; Nogura, Tsugunori[EN] Every selection f for the family F2(X) of at most two-point subsets of a set X naturally defines an order-like relation on X by if and only . In the present paper we study the relationship between and the possible topologies T on X which realize the continuity of f with respect to the Vietoris topology on F2(X) generated by T.
- PublicationDirected GF-spaces(Universitat Politècnica de València, 2001-10-01) Arenas, F.G.; Sánchez Granero, M.A.; Ministerio de Ciencia y Tecnología[EN] In this paper we introduce the concept of directed fractal structure, which is a generalization of the concept of fractal structure (introduced by the authors). We study the relation with transitive quasiuniformities and inverse limits of posets. We define the concept of GF-compactification and apply it to prove that the Stone-Cech compactification can be obtained as the GF-compactification of the directed fractal structure associated to the Pervin quasi-uniformity.