Applied General Topology - Vol 09, No 1 (2008)
https://riunet.upv.es:443/handle/10251/85921
2019-07-23T18:01:03ZProduct metrics and boundedness
https://riunet.upv.es:443/handle/10251/85938
Product metrics and boundedness
Beer, Gerald
[EN] This paper looks at some possible ways of equipping a countable product of unbounded metric spaces with a metric that acknowledges the boundedness characteristics of the factors.
2017-07-28T08:41:59ZSymmetric Bombay topology
https://riunet.upv.es:443/handle/10251/85937
Symmetric Bombay topology
Di Maio, Giuseppe; Meccariello, Enrico; Naimpally, Somashekhar
[EN] The subject of hyperspace topologies on closed or closed and compact subsets of a topological space X began in the early part of the last century with the discoveries of Hausdorff metric and Vietoris hit-and-miss topology. In course of time, several hyperspace topologies were discovered either for solving some problems in Applied or Pure Mathematics or as natural generalizations of the existing ones. Each hyperspace topology can be split into a lower and an upper part. In the upper part the original set inclusion of Vietoris was generalized to proximal set inclusion. Then the topologization of the Wijsman topology led to the upper Bombay topology which involves two proximities. In all these developments the lower topology, involving intersection of finitely many open sets, was generalized to locally finite families but intersection was left unchanged. Recently the authors studied symmetric proximal topology in which proximity was used for the first time in the lower part replacing intersection with its generalization: nearness. In this paper we use two proximities also in the lower part and we obtain the lower Bombay hypertopology. Consequently, a new hypertopology arises in a natural way: the symmetric Bombay topology which is the join of a lower and an upper Bombay topology.
2017-07-28T08:38:15ZFunctorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics
https://riunet.upv.es:443/handle/10251/85936
Functorial comparisons of bitopology with topology and the case for redundancy of bitopology in lattice-valued mathematics
Rodabaugh, S.E.
[EN] This paper studies various functors between (lattice-valued) topology and (lattice-valued) bitopology, including the expected “doubling” functor Ed : L-Top → L-BiTop and the “cross” functor E× : L-BiTop → L2-Top introduced in this paper, both of which are extremely well-behaved strict, concrete, full embeddings. Given the greater simplicity of lattice-valued topology vis-a-vis lattice-valued bitopology and the fact that the class of L2-Top’s is strictly smaller than the class of L-Top’s encompassing fixed-basis topology, the class of E×’s makes the case that lattice-valued bitopology is categorically redundant. As a special application, traditional bitopology as represented by BiTop is (isomorphic in an extremely well-behaved way to) a strict subcategory of 4-Top, where 4 is the four element Boolean algebra; this makes the case that traditional bitopology is a special case of a much simpler fixed-basis topology.
2017-07-28T08:35:28ZThe Cech number of Cp(X) when X is an ordinal space
https://riunet.upv.es:443/handle/10251/85933
The Cech number of Cp(X) when X is an ordinal space
Alas, Ofelia T.; Tamariz-Mascarúa, Ángel
[EN] The Cech number of a space Z, C(Z), is the pseudocharacter of Z in βZ. In this article we obtain, in ZFC and assuming SCH, some upper and lower bounds of the Cech number of spaces Cp(X) of realvalued continuous functions defined on an ordinal space X with the pointwise convergence topology
2017-07-28T08:00:19ZContinuous extension in topological digital spaces
https://riunet.upv.es:443/handle/10251/85932
Continuous extension in topological digital spaces
Melin, Erik
[EN] We give necessary and sufficient conditions for the existence of a continuous extension from a smallest-neighborhood space (Alexandrov space) X to the Khalimsky line. Using this result, we classify the subsets A X such that every continuous function A ! Zbcan be extended to all of X. We also consider the more general case ofbmappings X ! Y between smallest-neighborhood spaces, and prove abdigital no-retraction theorem for the Khalimsky plane
2017-07-28T07:56:13ZLow separation axioms via the diagonal
https://riunet.upv.es:443/handle/10251/85931
Low separation axioms via the diagonal
Colasante, María Luisa; Uzcátegui, Carlos; Vielma, Jorge
[EN] In the context of a generalized topology g on a set X, we give in this article characterizations of some separation axioms between T0 and T2 in terms of properties of the diagonal in X × X.
2017-07-28T07:52:01ZFunction Spaces and Strong Variants of Continuity
https://riunet.upv.es:443/handle/10251/85930
Function Spaces and Strong Variants of Continuity
Kohli, J.K.; Singh, D.
[EN] It is shown that if domain is a sum connected space and range is a T0-space, then the notions of strong continuity, perfect continuity and cl-supercontinuity coincide. Further, it is proved that if X is a sum connected space and Y is Hausdorff, then the set of all strongly continuous (perfectly continuous, cl-supercontinuous) functions is closed in Y X in the topology of pointwise convergence. The results obtained in the process strengthen and extend certain results of Levine and Naimpally.
2017-07-28T07:49:10ZExponentiality for the construct of affine sets
https://riunet.upv.es:443/handle/10251/85929
Exponentiality for the construct of affine sets
Claes, Veerle
[EN] The topological construct SSET of affine sets over the two-point set S contains many interesting topological subconstructs such as TOP, the construct of topological spaces, and CL, the construct of closure spaces. For this category and its subconstructs cartesian closedness is studied. We first give a classification of the subconstructs of SSET according to their behaviour with respect to exponenttiality. We formulate sufficient conditions implying that a subconstruct behaves similar to CL. On the other hand, we characterize a conglomerate of subconstructs with behaviour similar to TOP. Finally, we construct the cartesian closed topological hull of SSET
2017-07-28T07:44:57ZCancellation of 3-Point Topological Spaces
https://riunet.upv.es:443/handle/10251/85928
Cancellation of 3-Point Topological Spaces
Carter, Sheila; Craveiro de Carvalho, F.J.
[EN] The cancellation problem, which goes back to S. Ulam, is formulated as follows:
Given topological spaces X, Y, Z, under what circumstances does X × Z ≈Y × Z (≈ meaning homeomorphic to) imply X ≈ Y ?
In it is proved that, for T0 topological spaces and denoting by S the Sierpinski space, if X × S≈Y × S then X≈Y.
This note concerns all nine (up to homeomorphism) 3-point spaces, which are given in.
2017-07-28T07:42:48ZTopological and categorical properties of binary trees
https://riunet.upv.es:443/handle/10251/85927
Topological and categorical properties of binary trees
Pajoohesh, H.
[EN] Binary trees are very useful tools in computer science for estimating the running time of so-called comparison based algorithms, algorithms in which every action is ultimately based on a prior comparison between two elements. For two given algorithms A and B where the decision tree of A is more balanced than that of B, it is known that the average and worst case times of A will be better than those of B, i.e., ₸A(n) ≤₸B(n) and TWA (n)≤TWB (n). Thus the most balanced and the most imbalanced binary trees play a main role. Here we consider them as semilattices and characterize the most balanced and the most imbalanced binary trees by topological and categorical properties. Also we define the composition of binary trees as a commutative binary operation, *, such that for binary trees A and B, A * B is the binary tree obtained by attaching a copy of B to any leaf of A. We show that (T,*) is a commutative po-monoid and investigate its properties.
2017-07-28T07:40:25Z