Some Observations on Ideal Variations of Bornological Covers

Subhankar Das

doi:10.2478/tmmp-2023-0020

.blurhash-client-img { display: none !important; }

Some Observations on Ideal Variations of Bornological Covers

Tatra Mountains Mathematical Publications

Volume 85 (2023): Issue 3 (November 2023)

By: Subhankar Das

Open Access

|Nov 2023

Abstract

In this article, we use the notion of ideals to study open covers and related selection principles, and thus, we extend some results in (Caserta et al. 2012; Chandra et al. 2020) where open covers and related selection principles have been investigated using the idea of strong uniform convergence (Beer and Levi, 2009) on a bornology. We introduce the notions of ℐ-γ_ℬ s -cover, ℐ-strong-ℬ-Hurewicz and ℐ-strong-ℬ-groupable cover. Also, in (C(X),τs_ℬ), some properties like ℐ-strictly Frèchet Urysohn, ℐ-Reznichenko property are investigated.

References

ARKHANGEL’SKIĬ, A. V.: Hurewicz spaces, analytic sets and fan tightness of function spaces,SovietMath. Dokl. 33 (1986), 396–399.
Search in Google Scholar Back to article
ARKHANGEL’SKIĬ, A. V.: Topological Function Spaces. (Translated from the Russian by R. A. M. Hoksbergen.) In: Mathematics and its Applications (Soviet Series), Vol. 78. Kluwer Academic Publishers Group, Dordrecht, 1992.
Search in Google Scholar Back to article
BEER, G.—LEVI, S.: Strong uniform continuity, J. Math. Anal. Appl. 350 (2009), 568–589.
Search in Google Scholar Back to article
CASERTA, A.—DI MAIO, G.—HOLÁ, L’: Arzelà’s theorem and strong uniform convergence on bornologies, J. Math. Anal. Appl. 371 (2010), 384–392.
Search in Google Scholar Back to article
CASERTA, A.—DI MAIO, G.—KOČINAC, LJ. D. R.: Bornologies, selection principles and function spaces, Topology Appl. 159 (2012), no. 7, 1847–1852.
Search in Google Scholar Back to article
CHANDRA,D.—DAS, P.—- DAS, S.: Applications of bornological covering properties in metric spaces, Indag. Math. 31 (2020), 43–63.
Search in Google Scholar Back to article
CHANDRA, D.—DAS, P.—DAS, S.: Certain observations on selection principles from (a) bornological viewpoint, Quaest. Math. 45 (2022), no. 3, 423–442.
Search in Google Scholar Back to article
DAS, P.: Certain types of open covers and selection principles using ideals, Houston J. Math. 39 (2013), no. 2, 637–650.
Search in Google Scholar Back to article
DAS, P.—KOČINAC, LJ. D. R.—CHANDRA, D.: Some remarks on open covers and selection principles using ideals, Topology Appl. 202 (2016), 183–193.
Search in Google Scholar Back to article
DAS, S.—CHANDRA, D.: Certain observations on statistical variations of bornological covers, Filomat. 35 (2021), no. 7, 2303–2315.
Search in Google Scholar Back to article
DI MAIO, G.—KOČINAC, LJ. D. R.: Statistical convergence in topology, Topology Appl. 156 (2008), 28–45.
Search in Google Scholar Back to article
ENGELKING, R.: General Topology (2nd edition). In: Sigma Ser. Pure Math. Vol. 6. Heldermann, Berlin, 1989.
Search in Google Scholar Back to article
FAST, H.: Sur la convergence statistique, Colloq. Math. 2 (1951), 241–244.
Search in Google Scholar Back to article
HOGBE-NLEND, H.: Bornologies and Functional Analysis. North-Holland, Amsterdam, 1977.
Search in Google Scholar Back to article
JUST, W.—MILLER, A. W.—SCHEEPERS, M.—SZEPTYCKI, P. J.: The combinatorics of open covers (II), Topology Appl. 73 (1996), 241–266.
Search in Google Scholar Back to article
KOČINAC, LJ. D. R.—SCHEEPERS, M.: Combinatorics of open covers (VII), Fund. Math. 179 (2003), 131–155.
Search in Google Scholar Back to article
KOČINAC, LJ. D. R.: Selected results on selection principles. In: Proc. Third Seminar on Geometry and Topology, Tabriz, Iran, July 15-–17, (2004), pp. 71–104.
Search in Google Scholar Back to article
KOČINAC, LJ. D. R.: Selection principles related to α_i -properties, Taiwanese J. Math. 12 (2008), 561–571.
Search in Google Scholar Back to article
KOSTYRKO, P.—ŠALÁT, T,—WILCZYŃSKI, W.: ℐ-convergence, Real Anal. Exchange 26 (2000/2001), 669–686.
Search in Google Scholar Back to article
LAHIRI, B. K.—DAS, P.: ℐ and ℐ^∗-convergence in topological spaces, Math. Bohem. 130 (2005), 153–160.
Search in Google Scholar Back to article
LAHIRI, B. K.—DAS, P.: ℐ and ℐ^∗-convergence of nets, Real Anal. Exchange 32 (2008), 431–442.
Search in Google Scholar Back to article
MCCOY, R. A.—NTANTU, I.: Topological Properties of Spaces of Continuous Functions. In: Lecture Notes in Math. Vol. 1315, Springer-Verlag, Berlin, 1988.
Search in Google Scholar Back to article
SCHOENBERG, I. J.: The integrability of certain functions and related summability methods,Amer. Math. Monthly 66 (1959), 361–375.
Search in Google Scholar Back to article
SCHEEPERS, M.: Combinatorics of open covers (I): Ramsey theory, Topology Appl. 69 (1996), 31–62.
Search in Google Scholar Back to article
SCHEEPERS, M.: Selection principles and covering properties in topology, Note Mat. 22 (2003/2004), no. 2, 3–41.
Search in Google Scholar Back to article
TSABAN, B.: Some new directions in infinite combinatorial topology. In: (J. Bagaria, S. Todorcevic, eds.), Set theory. Centre de Recerca Matemàtica Barcelona, 2003–2004. In: Trends in Mathematics, Birkhauser, Basel 2006, pp. 225–255.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2023-0020 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

Journal RSS Feed

Language: English

Page range: 1 - 16

Submitted on: Jun 20, 2021

Published on: Nov 16, 2023

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

Bornology,

selection principles,

topology of strong uniform convergence,

ℐ- ℬ^s-Hurewicz property,

ℐ-Reznichenko property,

ℐ-strictly Frèchet Urysohn property

Related subjects:

Mathematics,

General mathematics

© 2023 Subhankar Das, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 85 (2023): Issue 3 (November 2023)