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Year 2025, Volume: 13 Issue: 2, 278 - 291, 31.10.2025

Necati Can Açıkgöz , Ceren Sultan Elmalı

Abstract

References

  • [1] N.C. Ac¸ıkg¨oz and C.S. Elmalı, On almost set-Menger spaces in bitopological context, AIMS Math., 7(2022), 20579-20593.
  • [2] N.C. Ac¸ıkg¨oz and C.S. Elmalı, Nearly Menger covering property via bitopological spaces, AIMS Math., 12(2024), 34042-34066.
  • [3] N.C. Ac¸ıkg¨oz and C.S. Elmalı, A note on a novel type of Alster covering property in a bitopological setting, Turk J Math, 49(2025), 185-200.
  • [4] N.C. Ac¸ıkg¨oz and C.S. Elmalı, A note on nearly Alster spaces and its interrelations with some covering properties, Filomat, accepted.
  • [5] G.K. Banerjee, On pairwise almost strongly q continuous mappings, Bull. Calcutta Math. Soc., 79(1987), 314–320.
  • [6] M. Bonanzinga, F. Cammaroto and Lj DR. Koˇcinac, Star Hurewicz and related properties, Appl. Gen. Topol., 5(2004), 79-89.
  • [7] M. Bonanzinga, F. Cammaroto, Lj DR. Koˇcinac and M.V. Matveev, On weaker forms Menger, Rothberger and Hurewicz properties, Mat. Vesn., 61(2009), 13-23.
  • [8] E. Borel, Sur la classification des ensembles de mesure nulle, Bull. Soc. Math. de France, 47(1919), 97-125.
  • [9] S. Bose and D. Sinha, Almost open, almost closed, q continuous, almost quasi compact mappings in bitopological spaces, Bull. Calcutta Math. Soc., 73(1981), 345-354.
  • [10] F. Cammaroto and G. Santoro, Some counterexamples and properties on generalizations of Lindel¨of spaces, Internat. J. Math. & Math. Sci., 19(1996), 737-746.
  • [11] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topol. Appl., 29(1988), 93-106.
  • [12] M.C. Datta, Projective bitopological spaces II., J. Aust. Math. Soc., 13(1972), 119-128.
  • [13] B.P. Dvalishvili, Bitopological spaces: Theory, Relations with Generalized Algebraic Structures, and Applications, North-Holland Mathematic Studies, Amsterdam, 2005.
  • [14] R. Engelking, General Topology. Heldermann Verlag, Berlin, 1989.
  • [15] A.E. Eysen and S. O¨ zc¸ag˘, Weaker forms of the Menger property in bitopological spaces, Quaest. Math., 41(2018), 877-888.
  • [16] A.E. Eysen and S. O¨ zc¸ag˘, Investigations on weak versions of the Alster property in bitopological spaces and selection principles, Filomat, 33(2019), 4561-457.
  • [17] A.E. Eysen, Weak and relative forms of some selection properties, Dissertation, Hacettepe University, 2019.
  • [18] W. Hurewicz, U¨ ber eine verallgemeinerung des Borelschen Theorems, Math. Z., 24(1926), 401-421.
  • [19] M. Jeli´c, Feebly p-continuous mappings, Suppl. Rend. Circ. Mat. Palermo, 24(1990), 387-395.
  • [20] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics of open covers II, Topol. Appl., 73(1996), 241-266.
  • [21] J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc., 13(1963), 71-89.
  • [22] M.D. Khan and A. Sabah, Selection principles and covering properties in bitopological spaces, Appl. Gen. Topol., 21(2020), 159-169.
  • [23] F.H. Khedr and A.M. Alshibani, On pairwise super continuous mappings in bitopological spaces, Internat. J. Math. & Math. Sci., 14(1991), 715-722.
  • [24] A. Kılıc¸man and Z. Salleh, On pairwise weakly Lindel¨of bitopological spaces, Bull. Iran. Math. Soc., 39(2013), 469-486.
  • [25] A. Kılıc¸man and Z. Salleh, Product properties for pairwise Lindel¨of spaces, Bull. Malays. Math. Sci. Soc., 34(2011), 231-246.
  • [26] A. Kılıc¸man and Z. Salleh, Pairwise Almost Lindel¨of bitopological spaces II, Malays. J. Math. Sci., 1(2007), 227-238.
  • [27] A. Kılıc¸man and Z. Salleh, Pairwise weakly regular-Lindel¨of spaces, Abstr. Appl. Anal., Article ID 184243, (2008), 13 pages.
  • [28] D. Kocev, Almost Menger and related spaces, Mat. Vesn., 61(2009), 173-180.
  • [29] Lj DR. Koˇcinac, Star Menger and related spaces, II, Filomat, 13(1999), 129-140.
  • [30] Lj DR. Koˇcinac, Generalized open sets and selection properties, Filomat, 33(2019), 1485-1493.
  • [31] Lj DR. Koˇcinac, Star Menger and related spaces, Publ. Math. Debrecen, 55(1999), 421-431.
  • [32] Lj DR. Koˇcinac, Selected results on selection principles. In: Proc. Third. Sem. Geometry and Topology, Tabriz, Iran, 2004, 71-104.
  • [33] Lj DR. Koˇcinac, Variations of classical selection principles: An overview, Quaest. Math., 43(2020), 1121-1153.
  • [34] Lj DR. Koˇcinac, S¸ . Konca and S. Singh, Set star-Menger and set strongly star-Menger spaces, Math. Slovaca, 72(2022), 185-196.
  • [35] Lj DR. Kocˇinac and S. O¨ zc¸ag˘, Bitopological spaces and selection principles. In: Proc ICTA 2011, Islamabad, Pakistan, 2012, 243-255.
  • [36] Lj DR. Kocˇinac and S. O¨ zc¸ag˘, More on selective covering properties in bitopological spaces, J. Math., Article ID: 5558456, (2021), 9 pages.
  • [37] S¸ . Konca, Weaker forms of some star selection properties, Konuralp J. Math., 9(2021), 245-249.
  • [38] R. Lakehal, Lj DR. Koˇcinac and D. Seba, Neighbourhood star selection properties in bitopological spaces, Filomat, 35(2019), 339-351.
  • [39] D. Lyakhovets and A.V. Osipov, Selection principles and games in bitopological function spaces, Filomat, 33(2019), 4535-4540.
  • [40] S.N. Maheshwari and R. Prasad, Semi open sets and semi continuous function in bitopological spaces, Math Notae, 26(1977), 29-37.
  • [41] K. Menger, Einige U¨ berdeckungssa¨tze der Punktmengenlehren, Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien), 133(1924), 421-444.
  • [42] A.V. Osipov and S. O¨ zc¸ag˘, Variations of selective separability and tightness in function spaces with set-open topologies, Topol. Appl., 217(2017), 38-50.
  • [43] S. O¨ zc¸ag˘ and A.E. Eysen, Almost Menger property in bitopological spaces, Ukr. Math. J., 68(2016), 950-958.
  • [44] B.A. Pansera, Weaker forms of the Menger property, Quaest. Math., 35(2012), 161-169.
  • [45] A. Parvez and M.D. Khan, On nearly Menger and nearly star-Menger spaces, Filomat, 33(2019), 6219–6227.
  • [46] T.G. Raghavan and I.L. Reilly, Metrizability of quasi-metric spaces, J. Lond. Math. Soc., 15(1977), 169-172.
  • [47] F. Rothberger, Eine Versch¨arfung der Eigenschaft C, Fund. Math., 30(1938), 50-55.
  • [48] M. Sakai, Star versions of the Menger property, Topol. Appl., 176(2014), 22-34.
  • [49] Z. Salleh and A. Kılıc¸man, On pairwise nearly Lindel¨of bitopological spaces, Far East Journal of Mathematical Sciences, 77(2013), 147-171.
  • [50] Z. Salleh, M. Nurillaev, CMIC. Taib, Product Properties for Generalized Pairwise Lindel¨of Spaces, Mathematics and Statistics, 11(2023), 606-614.
  • [51] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topol. Appl., 69(1996), 31-62.
  • [52] M. Scheepers, Selection principles and covering properties in topology, Note Mat., 22(2003), 3-41.
  • [53] M. Scheepers, Lusin sets, Proc. Amer. Math. Soc., 127(1999), 251-257.
  • [54] A.R. Singal and S.P. Arya, On pairwise almost regular spaces, Glasnik Math., 6(1971), 335-343.
  • [55] M.K. Singal and A.R. Singal, Some more separation axioms in bitopological spaces, Ann. Son. Sci. Bruxelles, 84(1970), 207-230.
  • [56] Y.K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Ins. Math., 98(2015), 193-198.
  • [57] Y.K. Song, Remarks on almost Rothberger spaces and weakly Rothberger spaces, Quaest. Math., 38(2015), 317-325. [58] L.A. Steen and J.A. Seebach, Counterexamples in Topology, Dover Publ., New York, 1995.

More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces

Year 2025, Volume: 13 Issue: 2, 278 - 291, 31.10.2025

Necati Can Açıkgöz , Ceren Sultan Elmalı

Abstract

In this paper, we investigate some weaker forms of the Rothberger covering property in a bitopological setting. We investigate the relationships between the bitopological weakly Rothberger, bitopological almost Rothberger and the Rothberger covering property in classical topological spaces. We also consider them under certain type of mappings, subspaces and products. Later, we introduce new types of the star-Menger covering property in bitopological spaces. By giving counter-examples, we see that our new forms are different from some of the existing covering properties. We establish conditions for their equivalences and pose some of the related covering properties for future research.

Keywords

star-Lindelöf spaces bitopological spaces covering properties selection principles star-Menger spaces

References

  • [1] N.C. Ac¸ıkg¨oz and C.S. Elmalı, On almost set-Menger spaces in bitopological context, AIMS Math., 7(2022), 20579-20593.
  • [2] N.C. Ac¸ıkg¨oz and C.S. Elmalı, Nearly Menger covering property via bitopological spaces, AIMS Math., 12(2024), 34042-34066.
  • [3] N.C. Ac¸ıkg¨oz and C.S. Elmalı, A note on a novel type of Alster covering property in a bitopological setting, Turk J Math, 49(2025), 185-200.
  • [4] N.C. Ac¸ıkg¨oz and C.S. Elmalı, A note on nearly Alster spaces and its interrelations with some covering properties, Filomat, accepted.
  • [5] G.K. Banerjee, On pairwise almost strongly q continuous mappings, Bull. Calcutta Math. Soc., 79(1987), 314–320.
  • [6] M. Bonanzinga, F. Cammaroto and Lj DR. Koˇcinac, Star Hurewicz and related properties, Appl. Gen. Topol., 5(2004), 79-89.
  • [7] M. Bonanzinga, F. Cammaroto, Lj DR. Koˇcinac and M.V. Matveev, On weaker forms Menger, Rothberger and Hurewicz properties, Mat. Vesn., 61(2009), 13-23.
  • [8] E. Borel, Sur la classification des ensembles de mesure nulle, Bull. Soc. Math. de France, 47(1919), 97-125.
  • [9] S. Bose and D. Sinha, Almost open, almost closed, q continuous, almost quasi compact mappings in bitopological spaces, Bull. Calcutta Math. Soc., 73(1981), 345-354.
  • [10] F. Cammaroto and G. Santoro, Some counterexamples and properties on generalizations of Lindel¨of spaces, Internat. J. Math. & Math. Sci., 19(1996), 737-746.
  • [11] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topol. Appl., 29(1988), 93-106.
  • [12] M.C. Datta, Projective bitopological spaces II., J. Aust. Math. Soc., 13(1972), 119-128.
  • [13] B.P. Dvalishvili, Bitopological spaces: Theory, Relations with Generalized Algebraic Structures, and Applications, North-Holland Mathematic Studies, Amsterdam, 2005.
  • [14] R. Engelking, General Topology. Heldermann Verlag, Berlin, 1989.
  • [15] A.E. Eysen and S. O¨ zc¸ag˘, Weaker forms of the Menger property in bitopological spaces, Quaest. Math., 41(2018), 877-888.
  • [16] A.E. Eysen and S. O¨ zc¸ag˘, Investigations on weak versions of the Alster property in bitopological spaces and selection principles, Filomat, 33(2019), 4561-457.
  • [17] A.E. Eysen, Weak and relative forms of some selection properties, Dissertation, Hacettepe University, 2019.
  • [18] W. Hurewicz, U¨ ber eine verallgemeinerung des Borelschen Theorems, Math. Z., 24(1926), 401-421.
  • [19] M. Jeli´c, Feebly p-continuous mappings, Suppl. Rend. Circ. Mat. Palermo, 24(1990), 387-395.
  • [20] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics of open covers II, Topol. Appl., 73(1996), 241-266.
  • [21] J.C. Kelly, Bitopological spaces, Proc. Lond. Math. Soc., 13(1963), 71-89.
  • [22] M.D. Khan and A. Sabah, Selection principles and covering properties in bitopological spaces, Appl. Gen. Topol., 21(2020), 159-169.
  • [23] F.H. Khedr and A.M. Alshibani, On pairwise super continuous mappings in bitopological spaces, Internat. J. Math. & Math. Sci., 14(1991), 715-722.
  • [24] A. Kılıc¸man and Z. Salleh, On pairwise weakly Lindel¨of bitopological spaces, Bull. Iran. Math. Soc., 39(2013), 469-486.
  • [25] A. Kılıc¸man and Z. Salleh, Product properties for pairwise Lindel¨of spaces, Bull. Malays. Math. Sci. Soc., 34(2011), 231-246.
  • [26] A. Kılıc¸man and Z. Salleh, Pairwise Almost Lindel¨of bitopological spaces II, Malays. J. Math. Sci., 1(2007), 227-238.
  • [27] A. Kılıc¸man and Z. Salleh, Pairwise weakly regular-Lindel¨of spaces, Abstr. Appl. Anal., Article ID 184243, (2008), 13 pages.
  • [28] D. Kocev, Almost Menger and related spaces, Mat. Vesn., 61(2009), 173-180.
  • [29] Lj DR. Koˇcinac, Star Menger and related spaces, II, Filomat, 13(1999), 129-140.
  • [30] Lj DR. Koˇcinac, Generalized open sets and selection properties, Filomat, 33(2019), 1485-1493.
  • [31] Lj DR. Koˇcinac, Star Menger and related spaces, Publ. Math. Debrecen, 55(1999), 421-431.
  • [32] Lj DR. Koˇcinac, Selected results on selection principles. In: Proc. Third. Sem. Geometry and Topology, Tabriz, Iran, 2004, 71-104.
  • [33] Lj DR. Koˇcinac, Variations of classical selection principles: An overview, Quaest. Math., 43(2020), 1121-1153.
  • [34] Lj DR. Koˇcinac, S¸ . Konca and S. Singh, Set star-Menger and set strongly star-Menger spaces, Math. Slovaca, 72(2022), 185-196.
  • [35] Lj DR. Kocˇinac and S. O¨ zc¸ag˘, Bitopological spaces and selection principles. In: Proc ICTA 2011, Islamabad, Pakistan, 2012, 243-255.
  • [36] Lj DR. Kocˇinac and S. O¨ zc¸ag˘, More on selective covering properties in bitopological spaces, J. Math., Article ID: 5558456, (2021), 9 pages.
  • [37] S¸ . Konca, Weaker forms of some star selection properties, Konuralp J. Math., 9(2021), 245-249.
  • [38] R. Lakehal, Lj DR. Koˇcinac and D. Seba, Neighbourhood star selection properties in bitopological spaces, Filomat, 35(2019), 339-351.
  • [39] D. Lyakhovets and A.V. Osipov, Selection principles and games in bitopological function spaces, Filomat, 33(2019), 4535-4540.
  • [40] S.N. Maheshwari and R. Prasad, Semi open sets and semi continuous function in bitopological spaces, Math Notae, 26(1977), 29-37.
  • [41] K. Menger, Einige U¨ berdeckungssa¨tze der Punktmengenlehren, Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien), 133(1924), 421-444.
  • [42] A.V. Osipov and S. O¨ zc¸ag˘, Variations of selective separability and tightness in function spaces with set-open topologies, Topol. Appl., 217(2017), 38-50.
  • [43] S. O¨ zc¸ag˘ and A.E. Eysen, Almost Menger property in bitopological spaces, Ukr. Math. J., 68(2016), 950-958.
  • [44] B.A. Pansera, Weaker forms of the Menger property, Quaest. Math., 35(2012), 161-169.
  • [45] A. Parvez and M.D. Khan, On nearly Menger and nearly star-Menger spaces, Filomat, 33(2019), 6219–6227.
  • [46] T.G. Raghavan and I.L. Reilly, Metrizability of quasi-metric spaces, J. Lond. Math. Soc., 15(1977), 169-172.
  • [47] F. Rothberger, Eine Versch¨arfung der Eigenschaft C, Fund. Math., 30(1938), 50-55.
  • [48] M. Sakai, Star versions of the Menger property, Topol. Appl., 176(2014), 22-34.
  • [49] Z. Salleh and A. Kılıc¸man, On pairwise nearly Lindel¨of bitopological spaces, Far East Journal of Mathematical Sciences, 77(2013), 147-171.
  • [50] Z. Salleh, M. Nurillaev, CMIC. Taib, Product Properties for Generalized Pairwise Lindel¨of Spaces, Mathematics and Statistics, 11(2023), 606-614.
  • [51] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topol. Appl., 69(1996), 31-62.
  • [52] M. Scheepers, Selection principles and covering properties in topology, Note Mat., 22(2003), 3-41.
  • [53] M. Scheepers, Lusin sets, Proc. Amer. Math. Soc., 127(1999), 251-257.
  • [54] A.R. Singal and S.P. Arya, On pairwise almost regular spaces, Glasnik Math., 6(1971), 335-343.
  • [55] M.K. Singal and A.R. Singal, Some more separation axioms in bitopological spaces, Ann. Son. Sci. Bruxelles, 84(1970), 207-230.
  • [56] Y.K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Ins. Math., 98(2015), 193-198.
  • [57] Y.K. Song, Remarks on almost Rothberger spaces and weakly Rothberger spaces, Quaest. Math., 38(2015), 317-325. [58] L.A. Steen and J.A. Seebach, Counterexamples in Topology, Dover Publ., New York, 1995.
There are 57 citations in total.

Details

Primary Language English
Subjects Mathematical Methods and Special Functions
Journal Section Research Article
Authors

Necati Can Açıkgöz

Ceren Sultan Elmalı

Publication Date October 31, 2025
Submission Date December 6, 2024
Acceptance Date April 28, 2025
Published in Issue Year 2025 Volume: 13 Issue: 2

Cite

APA Açıkgöz, N. C., & Elmalı, C. S. (2025). More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces. Konuralp Journal of Mathematics, 13(2), 278-291.
AMA Açıkgöz NC, Elmalı CS. More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces. Konuralp J. Math. October 2025;13(2):278-291.
Chicago Açıkgöz, Necati Can, and Ceren Sultan Elmalı. “More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces”. Konuralp Journal of Mathematics 13, no. 2 (October 2025): 278-91.
EndNote Açıkgöz NC, Elmalı CS (October 1, 2025) More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces. Konuralp Journal of Mathematics 13 2 278–291.
IEEE N. C. Açıkgöz and C. S. Elmalı, “More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces”, Konuralp J. Math., vol. 13, no. 2, pp. 278–291, 2025.
ISNAD Açıkgöz, Necati Can - Elmalı, Ceren Sultan. “More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces”. Konuralp Journal of Mathematics 13/2 (October2025), 278-291.
JAMA Açıkgöz NC, Elmalı CS. More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces. Konuralp J. Math. 2025;13:278–291.
MLA Açıkgöz, Necati Can and Ceren Sultan Elmalı. “More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces”. Konuralp Journal of Mathematics, vol. 13, no. 2, 2025, pp. 278-91.
Vancouver Açıkgöz NC, Elmalı CS. More on Rothberger Type Covering Properties and Novel Concepts for the Star-Menger Type Covering Properties in Bitopological Spaces. Konuralp J. Math. 2025;13(2):278-91.
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