On almost set-Menger spaces in bitopological context

Necati Can Açıkgöz; Ceren Sultan Elmalı; Necati Can Açıkgöz; Ceren Sultan Elmalı

doi:10.3934/math.20221128

AIMS Mathematics

2022, Volume 7, Issue 12: 20579-20593. doi: 10.3934/math.20221128

Previous Article Next Article

Research article

On almost set-Menger spaces in bitopological context

Department of Mathematics, Erzurum Technical University, Yakutiye, 25050, Turkey

Received: 01 June 2022 Revised: 26 August 2022 Accepted: 05 September 2022 Published: 20 September 2022
MSC : 03E72, 54D20, 54E55

In this paper, we define the $ ij $-almost-set Menger ($ ij $-ASM) property in bitopological spaces. We put up some equivalences of $ ij $-almost-set Menger bitopological spaces and investigate the behaviours of such spaces under some different types of mappings. We later take the preservation of these properties under union, subspaces, products into consideration and give some related examples. We finally introduce the concept of $ ij $-almost $ P_{\gamma} $-set in bitopological spaces.
- selection principles,
- almost Menger property,
- set Menger,
- bitopological spaces,
- covering properties
Citation: Necati Can Açıkgöz, Ceren Sultan Elmalı. On almost set-Menger spaces in bitopological context[J]. AIMS Mathematics, 2022, 7(12): 20579-20593. doi: 10.3934/math.20221128

Related Papers:

Abstract

In this paper, we define the $ ij $-almost-set Menger ($ ij $-ASM) property in bitopological spaces. We put up some equivalences of $ ij $-almost-set Menger bitopological spaces and investigate the behaviours of such spaces under some different types of mappings. We later take the preservation of these properties under union, subspaces, products into consideration and give some related examples. We finally introduce the concept of $ ij $-almost $ P_{\gamma} $-set in bitopological spaces.

References

[1]	W. Hurewicz, Über die Verallgemeinerung des Borelschen theorems, Math. Z., 24 (1926), 401–421.
[2]	W. Hurewicz, Über folgen stetiger funktionen, Fund. Math., 9 (1927), 193–204.
[3]	K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie, Wien), 133 (1924), 421–444.
[4]	F. Rothberger, Eine Verschärfung der Eigenschaft C, Fund. Math., 30 (1938), 50–55. https://doi.org/10.4064/fm-30-1-50-55 doi: 10.4064/fm-30-1-50-55
[5]	W. Just, A. W. Miller, M. Scheepers, P. J. Szeptycki, The combinatorics of open covers (Ⅱ), Topol. Appl., 73 (1996), 214–266. https://doi.org/10.1016/S0166-8641(96)00075-2 doi: 10.1016/S0166-8641(96)00075-2
[6]	M. Scheepers, Combinatorics of open covers (Ⅰ): Ramsey theory, Topol. Appl., 69 (1996), 195–202. https://doi.org/10.1016/0166-8641(95)00067-4 doi: 10.1016/0166-8641(95)00067-4
[7]	L. D. R. Kočinac, Selected results on selection principles, Proceedings of the Third Seminar on Geometry and Topology, Tabriz, Iran, 2004, 71–104.
[8]	M. Scheepers, Selection principles and covering properties in topology, Note Mat., 22 (2003/2004), 3–41.
[9]	B. Tsaban, Some new directions in infinite-combinatorial topology, Set theory (eds. J. Bagaria, S. Todorčevič), Birkhäuser, 2006,225–255. https://doi.org/10.1007/3-7643-7692-9
[10]	T. Banakh, D. Repovs, Universal nowhere dense and meager sets in Menger manifolds, Topol. Appl., 161 (2014), 127–140. https://doi.org/10.1016/j.topol.2013.09.012 doi: 10.1016/j.topol.2013.09.012
[11]	D. Repovs, L. Zdomskyy, S. Zhang, Countable dense homogeneous filters and the Menger covering property, Fund. Math., 224 (2014), 233–240. https://doi.org/10.4064/fm224-3-3 doi: 10.4064/fm224-3-3
[12]	M. Scheepers, Selection principles and Baire spaces, Mat. Vesn., 61 (2009), 195–202.
[13]	L. D. R. Kočinac, Star-Menger and related spaces, Ⅱ, Filomat, 13 (1999), 129–140.
[14]	L. D. R. Kočinac, Star-Menger and related spaces, Publ. Math. Debrecen, 55 (1999), 421–431.
[15]	D. Kocev, Almost Menger and related spaces, Mat. Vesn., 61 (2009), 173–180.
[16]	P. Daniels, Pixley-Roy spaces over subsets of the reals, Topol. Appl., 29 (1988), 93–106. https://doi.org/10.1016/0166-8641(88)90061-2 doi: 10.1016/0166-8641(88)90061-2
[17]	L. Babinkostova, B. A. Pansera, M. Scheepers, Weak covering properties and infinite games, Topol. Appl., 159 (2012), 3644–3657. https://doi.org/10.1016/j.topol.2012.09.009 doi: 10.1016/j.topol.2012.09.009
[18]	L. Babinkostova, B. A. Pansera, M. Scheepers, Weak covering properties and selection principles, Topol. Appl., 160 (2013), 2251–2271. https://doi.org/10.1016/j.topol.2013.07.022 doi: 10.1016/j.topol.2013.07.022
[19]	M. Bonanzinga, F. Cammaroto, B. A. Pansera, B. Tsaban, Diagonalizations of dense families, Topol. Appl., 165 (2014), 12–25. https://doi.org/10.1016/j.topol.2014.01.001 doi: 10.1016/j.topol.2014.01.001
[20]	D. Kocev, Menger-type covering properties of topological spaces, Filomat, 29 (2015), 99–106. https://doi.org/10.2298/FIL1501099K doi: 10.2298/FIL1501099K
[21]	G. Di Maio, L. D. R. Kočinac, A note on quasi-Menger and similar spaces, Topol. Appl., 179 (2015), 148–155.
[22]	B. A. Pansera, Weaker forms of the Menger property, Quaest. Math., 35 (2012), 161–169. https://doi.org/10.2989/16073606.2012.696830 doi: 10.2989/16073606.2012.696830
[23]	M. Sakai, Some weak covering properties and infinite games, Cent. Eur. J. Math., 12 (2014), 322–329. https://doi.org/10.2478/s11533-013-0343-4 doi: 10.2478/s11533-013-0343-4
[24]	Y. K. Song, Some remarks on almost Menger spaces and weakly Menger spaces, Publ. Inst. Math., 98 (2015), 193–198. https://doi.org/10.2298/PIM150513031S doi: 10.2298/PIM150513031S
[25]	A. V. Arhangel'skii, A generic theorem in the theory of cardinal invariants of topological spaces, Comment. Math. Univ. Carol., 36 (1995), 303–325.
[26]	A. V. Arhangel'skii, H. M. M. Genedi, Beginnings of the theory of relative topological properties, Gen. Topol., 1989, 3–48.
[27]	A. V. Arhangel'skii, Relative topological properties and relative topological spaces, Topol. Appl., 70 (1996), 87–99. https://doi.org/10.1016/0166-8641(95)00086-0 doi: 10.1016/0166-8641(95)00086-0
[28]	L. Babinkostova, Lj. D. R. Kočinac, M. Scheepers, Combinatorics of open covers (Ⅷ), Topol. Appl., 140 (2004), 15–32. https://doi.org/10.1016/j.topol.2003.08.019 doi: 10.1016/j.topol.2003.08.019
[29]	Lj. D. R. Kočinac, Ş. Konca, S. Singh, Variations of some star selection properties, AIP. Conf. Proc., 2334 (2021), 020006.
[30]	C. Guido, L. D. R. Kočinac, Relative covering properties, Quest. Answ. Gen. Topol., 19 (2001), 107–114.
[31]	L. D. R. Kočinac, C. Guido, L. Babinkostova, On relative $\gamma$-sets, East West J. Math., 2 (2000), 195–199.
[32]	L. D. R. Kočinac, Ş. Konca, Set-Menger and related properties, Topol. Appl., 275 (2020), 106996.
[33]	Ş. Konca, L. D. R. Kočinac, Set-star Menger and related spaces, Abstract Book Ⅵ ICRAPAM, İstanbul, Turkey, 2019, 49.
[34]	L. D. R. Kočinac, Ş. Konca, S. Singh, Set star-Menger and set strongly star-Menger spaces, Math. Slovaca, 72 (2022), 185–196.
[35]	L. D. R. Kočinac, S. Özçağ, Versions of separability in bitopological spaces, Topol. Appl., 158 (2011), 1471–1477.
[36]	L. D. R. Kočinac, S. Özçağ, Bitopological spaces and selection principles, Proceedings of International Conference on Topology and its Applications, Islamabad, Pakistan, 2011,243–255.
[37]	D. Lyakhovets, A. V. Osipov, Selection principles and games in bitopological function spaces, Filomat, 33 (2019), 4535–4540. https://doi.org/10.2298/FIL1914535L doi: 10.2298/FIL1914535L
[38]	A. V. Osipov, S. Özçağ, Variations of selective separability and tightness in function spaces with set-open topologies, Topol. Appl., 217 (2017), 38–50. https://doi.org/10.1016/j.topol.2016.12.010 doi: 10.1016/j.topol.2016.12.010
[39]	S. Özçağ, A. E. Eysen, Almost Menger property in bitopological spaces, Ukrainian Math. J., 68 (2016), 950–958.
[40]	A. E. Eysen, Investigations onweak versions of the Alster property in bitopological spaces and selection principles, Filomat, 33 (2019), 4561–4571. https://doi.org/10.2298/FIL1914561E doi: 10.2298/FIL1914561E
[41]	H. V. Chauhan, B. Singh, On almost Hurewicz property in bitopological spaces, Commun. Fac. Sci. Univ., 70 (2021) 74–81. https://doi.org/10.31801/cfsuasmas.710601 doi: 10.31801/cfsuasmas.710601
[42]	L. D. R. Kočinac, S. Özçağ, More on selective covering properties in bitopological spaces, J. Math., 2021, 5558456.
[43]	S. Özçağ, Introducing selective d-separability in bitopological spaces, Turk. J. Math., 46 (2022), 2109–2120.
[44]	R. Engelking, General topology, Heldermann-Verlag, 1989.
[45]	B. P. Dvalishvili, Bitopological spaces, theory, relations with generalized algebraic structures and applications, Elsevier Science B.V, 2005.
[46]	J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13 (1963), 71–89. https://doi.org/10.1112/plms/s3-13.1.71 doi: 10.1112/plms/s3-13.1.71
[47]	L. A. Steen, J. A. Seebach, Counterexamples in topology, Dover Publications, 1995.
[48]	A. B. Singal, S. P. Arya, On pairwise almost regular spaces, Glasnik Math., 26 (1971), 335–343.
[49]	F. H. Khedr, A. M. Alshibani, On pairwise super continuous mappings in bitopological spaces, Int. J. Math. Math. Sci., 14 (1991), 715–722. https://doi.org/10.1155/S0161171291000960 doi: 10.1155/S0161171291000960
[50]	A. Kılıçman, Z. Salleh, Pairwise almost lindelöf bitopological spaces Ⅱ, Malays J. Math. Sci., 1 (2007), 227–238.
[51]	L. Gillman, M. Henriksen, Concerning rings of continuous functions, Trans. Amer. Math. Soc., 77 (1954), 340–362. https://doi.org/10.1090/S0002-9947-1954-0063646-5 doi: 10.1090/S0002-9947-1954-0063646-5
[52]	Z. Salleh, Mappings and pairwise continuity on pairwise Lindelöf bitopological spaces, Albanian J. Math., 1 (2007), 115–120.
[53]	S. Bose, D. Sinha, Almost open, almost closed, $\theta$-continuous, almost quasi compact mappings in bitopological spaces, Bull. Calcutta Math. Soc., 73 (1981), 345–354.
[54]	M. C. Datta, Projective bitopological spaces Ⅱ, J. Aust. Math. Soc., 14 (1972), 119–128. https://doi.org/10.1017/S1446788700009708 doi: 10.1017/S1446788700009708
[55]	M. J. Saegrove, On bitopological spaces, Doctoral Dissertation, Iowa State University, Iowa, 1971.
[56]	A. E. Eysen, S. Özçağ, Weaker forms of the Menger property in bitopological spaces, Quaest. Math., 41 (2018), 877–888. https://doi.org/10.2989/16073606.2017.1415996 doi: 10.2989/16073606.2017.1415996
[57]	Z. Salleh, A. Kılıçman, Some results of pairwise almost Lindelöf spaces, JP J. Geometry Topol., 15 (2014), 81–98.
[58]	M. C. Datta, Projective bitopological spaces, J. Aust. Math. Soc., 13 (1972), 327–334. https://doi.org/10.1017/S1446788700013744 doi: 10.1017/S1446788700013744
[59]	J. Gerlits, Z. Nagy, Some properties of $C(X)$, I, Topol. Appl., 14 (1982), 151–161.
[60]	G. Di Maio, L. D. R. Kočinac, M. R. Žižović, Statistical convergence, selection principles and asymptotic analysis, Chaos Soliton. Fract., 42 (2009), 2815–2821.
[61]	G. Di Maio, L. D. R. Kočinac, Statistical convergence in topology, Topol. Appl., 156 (2008), 28–45.
[62]	S. Yadav, D. Gopal, P. Chaipunya, J. Martínez-Moreno, Towards strong convergence and Cauchy sequences in binary metric spaces, Axioms, 11 (2022), 383. https://doi.org/10.3390/axioms11080383 doi: 10.3390/axioms11080383
[63]	D. Gopal, P. Agarwal, P. Kumam, Metric structures and fixed point theory, CRC Press, 2021.
[64]	L. D. R. Kočinac, Selection properties in fuzzy metric spaces, Filomat, 26 (2012), 99–106.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)