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Keywords:
asymptotic density; statistical convergence; $\gamma$ cover; selection principle
asymptotic density; statistical convergence; $\gamma$ cover; selection principle
Summary:
The statistical convergence in a topological space is constrained in this study up to order $\alpha$, where $\alpha \in (0, 1)$. A fresh group of open covers (namely $s^{\alpha}$-$\gamma$ covers) and an entirely novel category of denseness (namely $s^{\alpha}$-denseness) are proposed using this notion of $s^{\alpha}$-convergence, which has been used to study various topological aspects of $s^{\alpha}$-density. It has been revealed that the class of $s^{\alpha}$-$\gamma$ coverings falls somewhere between the class of $\gamma$ covers and the class of $s$-$\gamma$ covers. The influence of $s^{\alpha}$-$\gamma$ covers in topological games and selection principles are also investigated.
References:
[1] Bal P.: A countable intersection like characterization of Star–Lindelöf spaces. Researches in Mathematics 31 (2023), no. 2, 3–7. DOI 10.15421/242308
[2] Bal P.: On strongly star $g$-compactness of topological spaces. Tatra Mt. Math. Publ. 85 (2023), 89–100.
[3] Bal P.: On the class of $I$-$\gamma$ open cover and $I$-$St$-$\gamma$ open covers. Hacettepe Journal of Mathematics and Statistics 52 (2023), no. 3, 630–639.
[4] Bal P., De R.: On strongly star semi-compactness of topological spaces. Khayyam J. Math. 9 (2023), no. 1, 54–60.
[5] Bal P., Kočinac L. D. R.: On selectively star-ccc spaces. Topology Appl. 281 (2020), Art. No. 107184, 8 pages.
[6] Bal P., Rakshit D.: A variation of the class of statistical $\gamma$ covers. Topol. Algebra Appl. 11 (2023), no. 1, 20230101, 9 pages.
[7] Bal P., Rakshit D., Sarkar S.: Countable compactness modulo an ideal of natural numbers. Ural Math. J. 9 (2023), no. 2, 28–35. DOI 10.15826/umj.2023.2.002
[8] Bhunia S., Das P., Pal S. K.: Restricting statistical convergence. Acta Math. Hungar. 134 (2012), no. 1–2, 153–161. DOI 10.1007/s10474-011-0122-2
[9] Çolak R.: Statistical convergence of order $\alpha$. Acta Mathematica Scientia 31 (2010), no. 3, 121–129; in Modern Methods in Analysis and Its Applications, Anamaya Publication, New Delhi, 2010, pages 121–129.
[10] Çolak R., Bektaş Ç. A.: $\lambda$-statistical convergence of order $\alpha$. Acta Math. Sci. Ser. B (Engl. Ed.) 31 (2011), no. 3, 953–959.
[11] Connor J. S.: The statistical and strong $p$-Cesàro convergence of sequences. Analysis 8 (1988), no. 1–2, 47–63. DOI 10.1524/anly.1988.8.12.47
[12] Das P.: Certain types of open covers and selection principles using ideals. Houston J. Math. 39 (2013), no. 2, 637–650.
[13] Engelking R.: General Topology. Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Zbl 0684.54001
[14] Fast H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241–244 (French). DOI 10.4064/cm-2-3-4-241-244 | Zbl 0044.33605
[15] Fridy J. A.: On statistical convergence. Analysis 5 (1985), no. 4, 301–313. DOI 10.1524/anly.1985.5.4.301 | Zbl 0588.40001
[16] Kočinac L. D. R.: Selected results on selection principles. in: Proc. of the 3rd Seminar on Geometry and Topology, Tabriz, 2004, pages 71–104.
[17] Kočinac L. D. R.: Selection principles related to $\alpha_i$-properties. Taiwanese J. Math. 12 (2008), no. 3, 561–571.
[18] Di Maio G., Kočinac L. D. R.: Statistical convergence in topology. Topology Appl. 156 (2008), no. 1, 28–45. DOI 10.1016/j.topol.2008.01.015 | Zbl 1155.54004
[19] Scheepers M.: Selection principles and covering properties in topology. Note Mat. 22 (2003/2004), no. 2, 3–41.
[20] Schoenberg I. J.: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361–375. DOI 10.1080/00029890.1959.11989303 | Zbl 0089.04002
[21] Zygmund A.: Trigonometrical Series. Monogr. Mat., 5, PWN-Państwowe Wydawnictwo Naukowe, Warszawa, 1935. Zbl 0065.05604
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