[1] M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mahmoud, $\beta $-open sets and $\beta $-continuous
mappings, Bull. Fac. Sci., Assiut Univ. 12, 77-90, 1983.
[2] M.E. Abd El-Monsef and A.M. Kozae, Some generalized forms of compactness and
closedness, Delta J. Sci. 9 (2), 257-269, 1985.
[3] D. Andrijević, Semi-preopen sets, Mat. Vesnik, 38, 24-32, 1986.
[4] Aqsa and M.U.D. Khan, On nearly Menger and nearly star Menger spaces, Filomat,
33, 6219-6227, 2019.
[5] M. Caldas and S. Jafari, A new decomposition of $\beta $-open functions, Chaos Solitons
Fractals 40, 10-12, 2009.
[6] D. Chodounsky, D. Repovs and L. Zdomskyy, Mathias forcing and combinatorial
covering properties of filters, J. Symb. Log. 80, 1398-1410, 2015.
[7] M. Ganster and D. Andrijević, On some questions concerning semi-preopen sets, J.
Inst. Math. & Comp. Sci. (Math. Ser.) 1 (2), 65-75, 1988.
[8] W. Hurewicz, Über die Verallgemeinerung des Borelschen Theorems, Math. Z. 24,
401-425, 1925.
[9] W. Hurewicz, Über Folgen stetiger Functionen, Fund. Math. 9, 193-204, 1927.
[10] L.D.R. Kocinac, The Pixley-Roy topology and selection principles, Questions Answers
Gen. Topology. 19, 219-225, 2001.
[11] L.D.R. Kocinac, Selected results on selection principles. Proceedings of the 3rd Semi-
nar on Geometry & Topology, 71-104, Azarb. Univ. Tarbiat Moallem, Tabriz, 2004.
[12] L.D.R. Kocinac, Generalized open sets and selection properties, Filomat, 33, 1485-
1493, 2019.
[13] L.D.R. Kocinac, Variations of classical selection principles: An overview, Quaest.
Math., 43 (8), 1121-1153, 2020.
[14] L.D.R. Kocinac, A. Sabah, M.U.D. Khan and D. Seba, Semi-Hurewicz spaces, Hacet.
J. Math. Stat. 46 (1), 53-66, 2017.
[15] L.D.R. Kocinac and M. Scheepers, Combinatorics of open covers (VII): Groupability,
Fund. Math. 179 (2), 131-155, 2003.
[16] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly 70, 36-41, 1963.
[17] A.S. Mashour, M.E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and weak
precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53, 47-53, 1982.
[18] K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt.
2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie,
Wien) 133, 421-444, 1924.
[19] A. Sabah, M.U.D. Khan and L.D.R. Ko˘cinac, Covering properties defined by semi-
open sets, J. Nonlinear Sci. Appl. 9 (6), 4388-4398, 2016.
[20] M. Sakai, The weak Hurewicz property of Pixley-Roy hyperspaces, Topology Appl.
160, 2531-2537, 2013.
[21] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69,
31-62, 1996.
[22] M. Scheepers, Selection principles and covering properties in topology, Note Mat. 22
(2), 3-41, 2003.
[23] L.A. Steen and J.A. Seebach, Counterexamples in Topology, Holt, Rinehart and Win-
ston, Inc., New York, 1970.
[24] B. Tsaban, Selection principles and special sets of reals, Open Problems in Topology
II (E. Pearl, editor), Elsevier Science Publishing, Amsterdam, 91–108, 2007.
[25] B.K. Tyagi, S. Singh and M. Bhardwaj, Covering properties defined by preopen sets,
Asian-European Journal of Mathematics, 14 (3), 2150035, 2021.
Recently, some papers on weaker forms of classical covering properties of Hurewicz and Menger have been published. In this paper, using the covers formed by $\beta $-open sets, we introduce and study the properties of $\beta $-Menger and $\beta $-Hurewicz topological spaces. We give counterexamples that show the interrelations between those properties. The subject of our investigation is also the preservation of $\beta $-Menger and $\beta $-Hurewicz properties under subspaces, products, and mappings.
[1] M.E. Abd El-Monsef, S.N. El-Deeb and R.A. Mahmoud, $\beta $-open sets and $\beta $-continuous
mappings, Bull. Fac. Sci., Assiut Univ. 12, 77-90, 1983.
[2] M.E. Abd El-Monsef and A.M. Kozae, Some generalized forms of compactness and
closedness, Delta J. Sci. 9 (2), 257-269, 1985.
[3] D. Andrijević, Semi-preopen sets, Mat. Vesnik, 38, 24-32, 1986.
[4] Aqsa and M.U.D. Khan, On nearly Menger and nearly star Menger spaces, Filomat,
33, 6219-6227, 2019.
[5] M. Caldas and S. Jafari, A new decomposition of $\beta $-open functions, Chaos Solitons
Fractals 40, 10-12, 2009.
[6] D. Chodounsky, D. Repovs and L. Zdomskyy, Mathias forcing and combinatorial
covering properties of filters, J. Symb. Log. 80, 1398-1410, 2015.
[7] M. Ganster and D. Andrijević, On some questions concerning semi-preopen sets, J.
Inst. Math. & Comp. Sci. (Math. Ser.) 1 (2), 65-75, 1988.
[8] W. Hurewicz, Über die Verallgemeinerung des Borelschen Theorems, Math. Z. 24,
401-425, 1925.
[9] W. Hurewicz, Über Folgen stetiger Functionen, Fund. Math. 9, 193-204, 1927.
[10] L.D.R. Kocinac, The Pixley-Roy topology and selection principles, Questions Answers
Gen. Topology. 19, 219-225, 2001.
[11] L.D.R. Kocinac, Selected results on selection principles. Proceedings of the 3rd Semi-
nar on Geometry & Topology, 71-104, Azarb. Univ. Tarbiat Moallem, Tabriz, 2004.
[12] L.D.R. Kocinac, Generalized open sets and selection properties, Filomat, 33, 1485-
1493, 2019.
[13] L.D.R. Kocinac, Variations of classical selection principles: An overview, Quaest.
Math., 43 (8), 1121-1153, 2020.
[14] L.D.R. Kocinac, A. Sabah, M.U.D. Khan and D. Seba, Semi-Hurewicz spaces, Hacet.
J. Math. Stat. 46 (1), 53-66, 2017.
[15] L.D.R. Kocinac and M. Scheepers, Combinatorics of open covers (VII): Groupability,
Fund. Math. 179 (2), 131-155, 2003.
[16] N. Levine, Semi-open sets and semi-continuity in topological spaces, Amer. Math.
Monthly 70, 36-41, 1963.
[17] A.S. Mashour, M.E. Abd El-Monsef and S.N. El-Deeb, On precontinuous and weak
precontinuous mappings, Proc. Math. Phys. Soc. Egypt 53, 47-53, 1982.
[18] K. Menger, Einige Überdeckungssätze der Punktmengenlehre, Sitzungsberichte Abt.
2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie,
Wien) 133, 421-444, 1924.
[19] A. Sabah, M.U.D. Khan and L.D.R. Ko˘cinac, Covering properties defined by semi-
open sets, J. Nonlinear Sci. Appl. 9 (6), 4388-4398, 2016.
[20] M. Sakai, The weak Hurewicz property of Pixley-Roy hyperspaces, Topology Appl.
160, 2531-2537, 2013.
[21] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69,
31-62, 1996.
[22] M. Scheepers, Selection principles and covering properties in topology, Note Mat. 22
(2), 3-41, 2003.
[23] L.A. Steen and J.A. Seebach, Counterexamples in Topology, Holt, Rinehart and Win-
ston, Inc., New York, 1970.
[24] B. Tsaban, Selection principles and special sets of reals, Open Problems in Topology
II (E. Pearl, editor), Elsevier Science Publishing, Amsterdam, 91–108, 2007.
[25] B.K. Tyagi, S. Singh and M. Bhardwaj, Covering properties defined by preopen sets,
Asian-European Journal of Mathematics, 14 (3), 2150035, 2021.
Kule, M. (2022). $\beta$-Menger and $\beta$-Hurewicz spaces. Hacettepe Journal of Mathematics and Statistics, 51(1), 1-7. https://doi.org/10.15672/hujms.911145
AMA
Kule M. $\beta$-Menger and $\beta$-Hurewicz spaces. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):1-7. doi:10.15672/hujms.911145
Chicago
Kule, Memet. “$\beta$-Menger and $\beta$-Hurewicz Spaces”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 1-7. https://doi.org/10.15672/hujms.911145.
EndNote
Kule M (February 1, 2022) $\beta$-Menger and $\beta$-Hurewicz spaces. Hacettepe Journal of Mathematics and Statistics 51 1 1–7.
IEEE
M. Kule, “$\beta$-Menger and $\beta$-Hurewicz spaces”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 1–7, 2022, doi: 10.15672/hujms.911145.
ISNAD
Kule, Memet. “$\beta$-Menger and $\beta$-Hurewicz Spaces”. Hacettepe Journal of Mathematics and Statistics 51/1 (February 2022), 1-7. https://doi.org/10.15672/hujms.911145.
JAMA
Kule M. $\beta$-Menger and $\beta$-Hurewicz spaces. Hacettepe Journal of Mathematics and Statistics. 2022;51:1–7.
MLA
Kule, Memet. “$\beta$-Menger and $\beta$-Hurewicz Spaces”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 1-7, doi:10.15672/hujms.911145.
Vancouver
Kule M. $\beta$-Menger and $\beta$-Hurewicz spaces. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):1-7.