Applied General Topology - Vol 03, No 1 (2002)
https://riunet.upv.es:443/handle/10251/82000
2024-09-09T07:27:44ZDuality and quasi-normability for complexity spaces
https://riunet.upv.es:443/handle/10251/82023
Duality and quasi-normability for complexity spaces
Romaguera, Salvador; Schellekens, M.P.
[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.
2017-05-30T10:57:55ZTopological groups with dense compactly generated subgroups
https://riunet.upv.es:443/handle/10251/82022
Topological groups with dense compactly generated subgroups
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.
2017-05-30T10:55:27ZFenestrations induced by perfect tilings
https://riunet.upv.es:443/handle/10251/82021
Fenestrations induced by perfect tilings
Arenas, F.G.; Puertas, M.L.
[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.
2017-05-30T10:52:56ZEvery finite system of T1 uniformities comes from a single distance structure
https://riunet.upv.es:443/handle/10251/82020
Every finite system of T1 uniformities comes from a single distance structure
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.
2017-05-30T10:50:47ZMinimal TUD spaces
https://riunet.upv.es:443/handle/10251/82019
Minimal TUD spaces
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.
2017-05-30T10:48:46ZAll hypertopologies are hit-and-miss
https://riunet.upv.es:443/handle/10251/82018
All hypertopologies are hit-and-miss
Naimpally, Somashekhar
[EN] We solve a long standing problem by showing that all known hypertopologies are hit-and-miss. Our solution is not merely of theoretical importance. This representation is useful in the study of comparison of the Hausdorff-Bourbaki or H-B uniform topologies and the Wijsman topologies among themselves and with others. Up to now some of these comparisons needed intricate manipulations. The H-B uniform topologies were the subject of intense activity in the 1960's in connection with the Isbell-Smith problem. We show that they are proximally locally finite topologies from which the solution to the above problem follows easily. It is known that the Wijsman topology on the hyperspace is the proximal ball (hit-and-miss) topology in”nice” metric spaces including the normed linear spaces. With the introduction of a new far-miss topology we show that the Wijsman topology is hit-and-miss for all metric spaces. From this follows a natural generalization of the Wijsman topology to the hyperspace of any T1 space. Several existing results in the literature are easy consequences of our work
2017-05-30T10:46:59ZOn paracompact spaces and projectively inductively closed functors
https://riunet.upv.es:443/handle/10251/82016
On paracompact spaces and projectively inductively closed functors
Zhuraev, T.F.
[EN] In this paper we introduce a notion of projectively inductively closed functor (p.i.c.-functor). We give sufficient conditions for a functor to be a p.i.c.-functor. In particular, any finitary normal functor is a p.i.c.-functor. We prove that every preserving weight p.i.c.- functor of a finite degree preserves the class of stratifiable spaces and the class of paracompact -spaces. The same is true (even if we omit a preservation of weight) for paracompact -spaces and paracompact p-spaces.
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2017-05-30T10:43:54ZSome results on best proximity pair theorems
https://riunet.upv.es:443/handle/10251/82015
Some results on best proximity pair theorems
Srinivasan, P.S.; Veeramani, P.
[EN] Best proximity pair theorems are considered to expound the sufficient conditions that ensure the existence of an element xo ϵ A, such that
d(xo; T xo) = d(A;B)
where T : A 2B is a multifunction defined on suitable subsets A and B of a normed linear space E. The purpose of this paper is to obtain best proximity pair theorems directly without using any multivalued fixed point theorem. In fact, the well known Kakutani's fixed point theorem is obtained as a corollary to the main result of this paper.
2017-05-30T10:41:34ZA contribution to fuzzy subspaces
https://riunet.upv.es:443/handle/10251/82014
A contribution to fuzzy subspaces
Alamar, Miguel; Estruch, Vicente D.
[EN] We give a new concept of fuzzy topological subspace, which extends the usual one, and study in it the related concepts of interior, closure and conectedness.
2017-05-30T10:39:15ZHausdorff compactifications and zero-one measures II
https://riunet.upv.es:443/handle/10251/82013
Hausdorff compactifications and zero-one measures II
Dimov, Georgi D.; Tironi, Gino
[EN] The notion of PBS-sublattice is introduced and, using it, a simplification of the results of [6] and of some results of [5] is obtained. Two propositions concerning Wallman-type compactifications are presented as well.
2017-05-30T10:33:25Z