Iterations and unions of star selection properties on topological spaces

Year 2025, Volume: 54 Issue: 1, 180 - 199, 28.02.2025

Javier Casas-de La Rosa , William Chen-mertens , Sergio A. Garcia-balan

https://doi.org/10.15672/hujms.1198061

Abstract

In this paper, we investigate what selection principles properties are possessed by small (with respect to the bounding and dominating numbers) unions of spaces with certain (star) selection principles. Furthermore, we give several results about iterations of these properties and weaker properties than paracompactness. In addition, we study the behaviour of these iterated properties on $\Psi$-spaces. Finally, we show that, consistently, there is a normal star-Menger space that is not strongly star-Menger; this example answers a couple of questions posed in [J. Casas-de la Rosa, S. A. Garcia-Balan, P. J. Szeptycki, Some star and strongly star selection principles, Topology Appl. 258, 572-587, 2019].

Keywords

Menger, star-Menger, strongly star-Menger, Hurewicz, star-Hurewicz, strongly star-Hurewicz, star selection principles, -spaces, iterated stars

Supporting Institution

Consejo Nacional de Ciencia y Tecnología (CONACYT, México)

Project Number

Scholarship 769010

Thanks

The first-listed author thanks to Consejo Nacional de Ciencia y Tecnología (CONACYT, México) for the financial support (Scholarship 769010) for this research.

References

• [1] M. Bonanzinga, Star-Lindelöf and absolutely star-Lindelöf spaces, Quest. Answ. Gen. Topol. 16, 79104, 1998.
• [2] M. Bonanzinga, F. Cammaroto, Lj.D.R. Kocinac, Star- Hurewicz and related properties, Appl. Gen.Topol. 5, 79-89, 2004.
• [3] M. Bonanzinga, F. Maesano, Selectively strongly star-Menger spaces and related properties, Atti della Accademia Peloritana dei Pericolanti, 99 (2), A2 2021.
• [4] M. Bonanzinga, M. Matveev, Some covering properties for -spaces, Mat. Vesn. 61, 3-11, 2009.
• [5] L. Bukovský, J. Haleš, On Hurewicz properties, Topology Appl. 132, 71-79, 2003.
• [6] D.K. Burke, Covering properties, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 347-422, 1984.
• [7] J. Casas-de la Rosa, S. A. Garcia-Balan, Variations of star selection principles on small spaces, Filomat 36 (14), 4903-4917, 2022.
• [8] J. Casas-de la Rosa, S. A. Garcia-Balan, P. J. Szeptycki, Some star and strongly star selection principles, Topology Appl. 258, 572-587, 2019.
• [9] D. Chandra, N. Alam, On certain star versions of the Scheepers property, arXiv: 2207.08595 [math.GN]
• [10] M. V. Cuzzupè, Some selective and monotone versions of covering properties and some results on the cardinality of a topological space, PhD thesis, University of Catania, 2017.
• [11] E.K. van Douwen, G.M. Reed, A.W. Roscoe, I.J. Tree, Star covering properties, Topology Appl. 39, 71-103, 1991.
• [12] R. Engelking, General Topology, Heldermann Verlag, Berlin, Sigma Series in Pure Mathematics 6, 1989.
• [13] S. A. Garcia-Balan, Results on star selection principles and weakenings of normality in $\Psi$-spaces, PhD thesis, York University, 2020.
• [14] F. Hernández-Hernández, M. Hrušák, Topology of Mrówka-Isbell spaces. In: Hrušák, Tamariz, Tkachenko (Eds.), Pseudocompact Topological Spaces, Springer International Publishing AG, 2018.
• [15] M. Hruák, O. Guzmán, n-Luzin gaps, Topology Appl. 160, 1364-1374, 2013.
• [16] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24 (1), 401-421, 1925.
• [17] Lj.D.R. Kocinac, Star-Menger and related spaces, Publ. Math. (Debr.) 55, 421-431, 1999.
• [18] N.N. Luzin, On subsets of the series of natural numbers, Izvestiya Akad. Nauk SSSR. Ser. Mat. 11, 403410, 1947.
• [19] Lj.D.R. Kocinac, Star selection principles: A survey, Khayyam J. Math. 1, 82-106, 2015.
• [20] M.V. Matveev, A survey on star covering properties, Topology Atlas, Preprint No. 330, 1998.
• [21] K.Menger, Einige überdeckungssätze der Punltmengen-lehre, Sitzungberichte Abt.2a, Mathematik, Astronomie, Physik, Meteorologie and Mechanik (Wiener Akademie, Wien) 133, 421-444, 1924.
• [22] D. Repovš, L. Zdomskyy, On the Menger covering property and D-spaces, Proc. Amer. Math. Soc. 140 (3), 10691074, 2012.
• [23] F. Rothberger, Eine Verschärfung der Eigenschaft C, Fund. Math. 30, 50-55, 1938.
• [24] M. E. Rudin, Dowker Spaces, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set- Theoretic Topology, North-Holland, Amsterdam, 761-780, 1984.
• [25] M. Scheepers, Combinatorics of open covers I: Ramsey Theory, Topology Appl. 69, 31-62, 1996.
• [26] Y.-K. Song, Remarks on countability and star covering properties, Topology Appl. 158, 1121-1123, 2011.
• [27] Y.-K. Song, Remarks on neighborhood star-Lindelöf spaces II, Filomat 27 (5), 875- 880, 2013.
• [28] Y.-K. Song, X.Wei-Feng, Remarks on new star-selection principles in topology, Topology Appl. 268, 106921, 2019.
• [29] F. D. Tall, Normality versus collectionwise normality, in: K. Kunen, J.E. Vaughan (Eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 685-732, 1984.
• [30] F. D. Tall, Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D, Topology Appl. 158 (18), 2556-2563, 2011.
• [31] Ian J. Tree, Constructing regular 2-starcompact spaces that are not strongly 2-star- Lindelöf, Topology Appl. 47, 129-132, 1992.