Shape as a Cantor completion process

Morón, Manuel A.; Portal, Francisco R. Ruiz del; Matematicas, U. D.

doi:10.1007/pl00004599

Cited by 12 publications

(10 citation statements)

References 14 publications

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“…Also Cuchillo-Ibanez et al [6] constructed many generalized ultrametrics in the set of shape morphisms between topological spaces and obtained semivaluations and valuations on the groups of shape equivalences and kth shape groups. On the other hands, Cuchillo-Ibanez et al [7] introduced a topology on the set Sh(X, Y ), where X and Y are arbitrary topological spaces, in such a way that it extended topologically the construction given in [19]. Also, Moszynska [21] showed that the kth shape groupπ k (X, x), k ∈ N, is isomorphic to the set Sh((S k , * ), (Xx)) consists of all shape morphisms (S k , * ) → (X, x) with a group operation.…”

Section: Introductionmentioning

confidence: 99%

On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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In this paper, using the topology on the set of shape morphisms between arbitrary topological spaces X, Y , Sh(X, Y ), defined by Cuchillo-Ibanez et al. in 1999, we consider a topology on the shape homotopy groups of arbitrary topological spaces which make them Hausdorff topological groups. We then exhibit an example in whichπ top k succeeds in distinguishing the shape type of X and Y whileπ k fails, for all k ∈ N. Moreover, we present some basic properties of topological shape homotopy groups, among them commutativity ofπ top k with finite product of compact Hausdorff spaces. Finally, we consider a quotient topology on the kth shape group induced by the kth shape loop space and show that it coincides with the above topology.

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Section: Introductionmentioning

confidence: 99%

On topological shape homotopy groups

Nasri

Ghanei

Mashayekhy

et al. 2016

Topology and its Applications

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“…During the 1990s we developed our program of equipping the sets of shape morphisms between spaces of useful structures in the sense that it could help us find new results in shape theory, as well as help to reinterpret known results in terms of such structures. See, in chronological order of conception, [18,19,10,20,21,9]. Although [9] was published recently, in fact it was presented as a talk, with the same title as the paper in the II Congreso Iberoamericano de Topología y sus aplicaciones hold at Morelia, Mexico, March 20-22, 1997.…”

Section: Introductionmentioning

confidence: 99%

“…Although [9] was published recently, in fact it was presented as a talk, with the same title as the paper in the II Congreso Iberoamericano de Topología y sus aplicaciones hold at Morelia, Mexico, March 20-22, 1997. In [18,19] we developed the compact metrizable case using the Hilbert cube as ambient space…”

Section: Introductionmentioning

confidence: 99%

Ultrametrics on Čech homology groups

Giraldo

Morón

Sánchez‐González

2019

Topology and its Applications

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To the memory of Sibe Mardesič to whom we will always be indebted MSC: 55N05 55P55 Keywords:Čech homology Čech homology groups Approximative cycle and boundary Approximative homology Ultrametric This paper is devoted to introducing additional structure on Čech homology groups. First, we redefine the Čech homology groups in terms of what we have called approximative homology by using approximative sequences of cycles, just as Borsuk introduced shape groups using approximative maps. From this point on, we are able to construct complete ultrametrics on Čech homology groups. The uniform type (and then the group topology) generated by the ultrametric leads to a shape invariant which we use to deduce topological consequences.

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“…This paper is mainly inspired by a recent, interesting and beautiful one due to Hughes [4] but it is also motivated by [8] where a complete ultrametric was defined on the sets of shape morphisms between compacta.…”

Section: Introductionmentioning

confidence: 99%

“…In [8] it was proved that every shape morphism induces a uniformly continuous map between the corresponding ultrametric spaces of shape morphisms which are, in particular, complete and bounded as metric spaces. Moreover Hughes established some categorical equivalences for some classes of ultrametric spaces and local similarity equivalences to certain categories of geodesically complete rooted R-trees and certain equivalence classes of isometries at infinity.…”

Section: Introductionmentioning

confidence: 99%