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Keywords:
density topology; ideal; $\mathcal {I}$-density topology
density topology; ideal; $\mathcal {I}$-density topology
Summary:
We introduce the notion of $\mathcal {I}_{(s)}$-density point corresponding to the family of unbounded and $\mathcal {I}$-monotonic increasing positive real sequences, where $\mathcal {I}$ is the ideal of subsets of the set of natural numbers. We study the corresponding topology in the space of reals and investigate several properties of this topology. Also we present a characterization of equality between the classical density topology and $\mathcal {I}_{(s)}$-density topology.
References:
[1] Banerjee, A. K.: Sparse set topology and the space of proximally continuous mappings. South Asian J. Math. 6 (2016), 58-63.
[2] Banerjee, A. K., Debnath, I.: On density topology using ideals in the space of reals. Filomat 38 (2024), 743-768. DOI 10.2298/FIL2402743B | MR 4674755
[3] Banerjee, A. K., Paul, A.: $I$-divergence and $I^*$-divergence in cone metric spaces. Asian-Eur. J. Math. 13 (2020), Article ID 2050139, 11 pages. DOI 10.1142/S1793557120501399 | MR 4153970 | Zbl 1462.54006
[4] Banerjee, A. K., Paul, A.: On $I$ and $I^*$-Cauchy conditions in $C^*$-algebra valued metric spaces. Korean J. Math. 29 (2021), 621-629. DOI 10.11568/kjm.2021.29.3.621 | MR 4337226 | Zbl 1494.54005
[5] Ciesielski, K., Larson, L., Ostaszewski, K.: ${\mathcal I}$-density continuous functions. Mem. Am. Math. Soc. 515 (1994), 133 pages. DOI 10.1090/memo/0515 | MR 1188595 | Zbl 0801.26001
[6] Das, P., Banerjee, A. K.: On the sparse set topology. Math. Slovaca 60 (2010), 319-326. DOI 10.2478/s12175-010-0014-x | MR 2646374 | Zbl 1265.28040
[7] Dasgupta, A.: Set Theory: With an Introduction to Real Point Sets. Birkhäuser, New York (2014). DOI 10.1007/978-1-4614-8854-5 | MR 3157351 | Zbl 1286.03001
[8] Fast, H.: Sur la convergence statistique. Colloq. Math. 2 (1951), 241-244 French. DOI 10.4064/cm-2-3-4-241-244 | MR 0048548 | Zbl 0044.33605
[9] Filipczak, M., Filipczak, T., Horbaczewska, G.: Density topologies for strictly positive Borel measures. Topology Appl. 303 (2021), Article ID 107857, 14 pages. DOI 10.1016/j.topol.2021.107857 | MR 4320431 | Zbl 1480.54006
[10] Filipczak, M., Hejduk, J.: On topologies associated with the Lebesgue measure. Tatra Mt. Math. Publ. 28 (2004), 187-197. MR 2086991 | Zbl 1107.54003
[11] Goffman, C., Waterman, D.: Approximately continuous transformations. Proc. Am. Math. Soc. 12 (1961), 116-121. DOI 10.1090/S0002-9939-1961-0120327-6 | MR 0120327 | Zbl 0096.17103
[12] Halmos, P. R.: Measure Theory. Graduate Texts in Mathematics 18. Springer, New York (1974). DOI 10.1007/978-1-4684-9440-2 | MR 0033869 | Zbl 0283.28001
[13] Hejduk, J., Lindner, S., Loranty, A.: On lower density type operators and topologies generated by them. Filomat 32 (2018), 4949-4957. DOI 10.2298/FIL1814949H | MR 3898543 | Zbl 1499.54032
[14] Hejduk, J., Loranty, A.: Remarks on the topologies in the Lebesgue measurable sets. Demonstr. Math. 45 (2012), 655-663. DOI 10.1515/dema-2013-0391 | MR 2987194 | Zbl 1280.54004
[15] Hejduk, J., Loranty, A.: On abstract and almost-abstract density topologies. Acta Math. Hung. 155 (2018), 228-240. DOI 10.1007/s10474-018-0838-3 | MR 3831294 | Zbl 1424.54009
[16] Hejduk, J., Wiertelak, R.: On the generalization of density topologies on the real line. Math. Slovaca 64 (2014), 1267-1276. DOI 10.2478/s12175-014-0274-y | MR 3277852 | Zbl 1349.54005
[17] Kostyrko, P., Šalát, T., Wilczyński, W.: $\mathcal I$-convergence. Real Anal. Exchange 26 (2000)/2001 669-686. MR 1844385 | Zbl 1021.40001
[18] Lahiri, B. K., Das, P.: Density topology in a metric space. J. Indian Math. Soc., New Ser. 65 (1998), 107-117. MR 1750361 | Zbl 1074.54517
[19] Lahiri, B. K., Das, P.: $I$ and $I^*$-convergence in topological spaces. Math. Bohem. 130 (2005), 153-160. DOI 10.21136/MB.2005.134133 | MR 2148648 | Zbl 1111.40001
[20] Oxtoby, J. C.: Measure and Category: A Survey of the Analogies Between Topological and Measure Spaces. Graduate Texts in Mathematics 2. Springer, Berlin (1980). DOI 10.1007/978-1-4684-9339-9 | MR 0584443 | Zbl 0435.28011
[21] Riesz, F.: Sur les points de densite au sens fort. Fundam. Math. 22 (1934), 221-225 French. DOI 10.4064/fm-22-1-221-225 | Zbl 0009.20802
[22] Šalát, T., Tripathy, B. C., Ziman, M.: On some properties of $\mathcal I$-convergence. Tatra Mt. Math. Publ. 28 (2004), 279-286. MR 2087000 | Zbl 1110.40002
[23] Schoenberg, I. J.: The integrability of certain functions and related summability methods. Am. Math. Mon. 66 (1959), 361-375. DOI 10.2307/2308747 | MR 0104946 | Zbl 0089.04002
[24] Strobin, F., Wiertelak, R.: On a generalization of density topologies on the real line. Topology Appl. 199 (2016), 1-16. DOI 10.1016/j.topol.2015.11.005 | MR 3442589 | Zbl 1332.54006
[25] Troyer, R. J., Ziemer, W. P.: Topologies generated by outer measures. J. Math. Mech. 12 (1963), 485-494. MR 0147600 | Zbl 0112.28401
[26] Wilczyński, W.: Density topologies. Handbook of Measure Theory North-Holland, Amsterdam (2002), 675-702. DOI 10.1016/B978-044450263-6/50016-6 | MR 1954625 | Zbl 1021.28002
[27] Wojdowski, W.: A generalization of the density topology. Real Anal. Exch. 32 (2006/2007), 349-358. DOI 10.14321/realanalexch.32.2.0349 | MR 2369849 | Zbl 1135.28001
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