Applied General Topology - Vol 02, No 2 (2001)
https://riunet.upv.es:443/handle/10251/81999
2019-11-18T09:53:56ZContinuous maps in the Bohr Topology
https://riunet.upv.es:443/handle/10251/82012
Continuous maps in the Bohr Topology
Dikranjan, Dikran
[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.
2017-05-30T10:24:35ZUnions of chains of subgroups of a topologucal group
https://riunet.upv.es:443/handle/10251/82010
Unions of chains of subgroups of a topologucal group
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.
2017-05-30T10:10:28ZOn strongly reflexive topological groups
https://riunet.upv.es:443/handle/10251/82009
On strongly reflexive topological groups
Chasco, M. J.; Martin-Peinador, E.
[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.
2017-05-30T10:08:29ZSelections and order-like relations
https://riunet.upv.es:443/handle/10251/82008
Selections and order-like relations
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.
2017-05-30T10:05:50ZDirected GF-spaces
https://riunet.upv.es:443/handle/10251/82007
Directed GF-spaces
Arenas, F.G.; Sánchez Granero, M.A.
[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.
2017-05-30T10:03:39ZGeneralized closed sets: a unified approach
https://riunet.upv.es:443/handle/10251/82006
Generalized closed sets: a unified approach
Cao, Jiling; Greenwood, Sina; Reilly, Ivan L.
[EN] We investigate various classes of generalized closed sets of a topological space in a unified way by studying the notion of qr-closed sets. New characterizations of some existing classes of generalized closed sets and topological spaces are given. A new class of generalized closed sets are introduced.
2017-05-30T10:00:11ZThe chainable continua are the spaces approximated by finite COTS
https://riunet.upv.es:443/handle/10251/82005
The chainable continua are the spaces approximated by finite COTS
Kennedy, Judy A.; Kopperman, Ralph D.; Wilson, Richard G.
[EN] We show that the chainable continua (also called snake-like or arc-like)continua, are precisely the Hausdorff reflections of inverse limits of sequences of finite COTS under maps which are continuous and are separating.
2017-05-30T09:58:01ZTopological normal forms of high degree for the simplest bifurcations
https://riunet.upv.es:443/handle/10251/82003
Topological normal forms of high degree for the simplest bifurcations
Balibrea, Francisco; Valverde Fajardo, Jose C.
[EN] This paper is devoted to study the topological normal forms of families of maps on R which, under nondegeneracy conditions of high degree, also present the simplest bifurcations.
2017-05-30T09:55:49ZCoreflectively modified continuous duality applied to classical product theorems
https://riunet.upv.es:443/handle/10251/82002
Coreflectively modified continuous duality applied to classical product theorems
Mynard, Frédéric
[EN] Several (old and new) results on stability of quotients (of various types) under product, on sequentiality of product of sequential spaces, on relationships between a topology and the upper Kuratowski convergence on its closed sets are derived from a general mechanism of duality that uses the continuous convergence lead to new reflectors which are of fundamental interest in this quest.
2017-05-30T09:52:38Z