Compactness in the endograph uniformity

Iván  Sánchez

doi:10.4995/agt.2024.19400

Authors

Iván Sánchez Universidad Autónoma Metropolitana https://orcid.org/0000-0002-4356-9147

DOI:

https://doi.org/10.4995/agt.2024.19400

Keywords:

Fuzzy sets, Endograph uniformity, Endograph metric, Sendograph uniformity, Sendograph metric, Completeness, Compactness

Abstract

Given a uniform space (X,U), we denote by F^*(X) to the family of fuzzy sets u in (X,U) such that u is normal and upper semicontinuous. Let U_E be the endograph uniformity on F^*(X). In this paper, we mainly characterize totally bounded and compact substes in the uniform space (F^*(X),U_E).

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Author Biography

Iván Sánchez, Universidad Autónoma Metropolitana

Departamento de Matemáticas

References

R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.

J. J. Font, D. Sanchis, and M. Sanchis, Completeness, metrizability and compactness in spaces of fuzzy-number-valued functions, Fuzzy Sets and Systems 353 (2018), 124-136. https://doi.org/10.1016/j.fss.2018.05.006

H. Huang, Characterizations of endograph metric and Γ-convergence on fuzzy, Fuzzy Sets and Systems 350 (2018), 55-85. https://doi.org/10.1016/j.fss.2018.02.014

H. Huang, Properties of several fuzzy set spaces, arXiv:1910.02205v2 [math.GM].

G. Huang, G.-B. Huang, S. Song, and K. You, Trends in extreme learning machines: a review, Neural Netw. 61 (2015), 32-48. https://doi.org/10.1016/j.neunet.2014.10.001

D. Jardón, I. Sánchez and M. Sanchis, Fuzzy sets on uniform spaces, preprint.

D. Jardón, I. Sánchez and M. Sanchis, Some questions about Zadeh's extension on metric spaces, Fuzzy Sets Syst. 379 (2020), 115-124. https://doi.org/10.1016/j.fss.2018.10.019

J. L. Kelley, General Topology, Springer, New York, NY, USA, 1975.

P. E. Kloeden, Compact supported endographs and fuzzy sets, Fuzzy Sets Syst. 4, no. 2 (1980), 193-201. https://doi.org/10.1016/0165-0114(80)90036-6

J. Kupka, On fuzzifications of discrete dynamical systems, Information Sciences 181 (2011), 2858-2872. https://doi.org/10.1016/j.ins.2011.02.024

E. Michael, Topologies on spaces of subsets, Transactions of the American Mathematical Society 71, no. 1 (1951), 152-182. https://doi.org/10.1090/S0002-9947-1951-0042109-4

K. Morita, Completion of hyperspaces of compact subsets and topological completion of open-closed maps, General Topology and its Applications (1974), 217-233. https://doi.org/10.1016/0016-660X(74)90023-3

M. Rojas-Medar, and H. Román-Flores, On the equivalence of convergences of fuzzy sets, Fuzzy Sets Syst. 80 (1996), 217-224. https://doi.org/10.1016/0165-0114(95)00182-4

F. Scarselli, and A. C. Tsoi, Universal approximation using feedforward neural networks: a survey of some existing methods, and some new results, Neural Netw. 11, no. 1 (1998), 15-37. https://doi.org/10.1016/S0893-6080(97)00097-X