Weakly strongly star-Menger spaces

  • Gaurav Kumar Department of Mathematics, University of Delhi, New Delhi-110007, India.
  • Brij K. Tyagi Atma Ram Sanatan Dharma College, University of Delhi, New Delhi-110021, India.
Keywords: Stronlgy star-Menger, star-Menger, almost star-Menger, Weakly star-Menger, covering topological spaces


A space \(X\) is called weakly strongly star-Menger space if for each sequence (\(\mathcal{U}_{n} : n \in \omega\)) of open covers of \(X\), there is a sequence \((F_n : n\in\omega)\) of finite subsets of \(X\) such that \(\overline{\bigcup_{n\in\omega} St(F_n, \mathcal{U}_n)}\) is \(X\). In this paper, we investigate the relationship of weakly strongly star-Menger spaces with other related spaces. It is shown that a Hausdorff paracompact weakly star Menger \(P\)-space is star-Menger. We also study the images and preimages of weakly strongly star-Menger spaces under various type of maps.


M. Bonanzinga, F. Cammaroto and Lj. D. R. Kočinac, “Star-Hurewicz and related properties”, Appl. Gen. Topol., vol. 5, no. 1, pp. 79-89, 2004.

M. Bonanzinga, F. Cammaroto, Lj. D. R. Kočinac and M. V. Matveev, “On weaker forms of Menger, Rothberger and Hurewicz properties”, Mat. Vesnik, vol 61, no. 1, pp. 13-23, 2019.

M. Bonanzinga and M. V. Matveev, “Some covering properties for ψ -spaces”, Mat. Vesnik, vol. 61, no. 1, pp. 3–11, 2009.

M. Bonanzinga, M. V. Matveev and B. A. Pansera, “When can a cover of a product be refined by a product of covers”, Question Answers Gen. Topology, vol. 26, no. 2, pp. 67-74, 2008.

M. Bonanzinga and B. A. Pansera, “Relative versions of some star selection principles”, Acta Math. Hungar., vol. 117, no. 3, pp. 231-243, 2007.

A. Caserta, G. M. Di Maio and Lj. D. R. Kočinac, “Versions of properties (a) and (pp)”, Topology Appl., vol. 158, no. 12, pp. 1360–1368, 2011.

E. K. van Douwen, G. K. Reed, A. W. Roscoe and I. J. Tree, “Star covering properties”, Topology Appl., vol. 39, no. 1, pp. 71–103, 1991.

E. K. van Douwen, “The integers and topology”, in: K. Kunen, J.E. Vaughan (Eds.), Hand- book of Set-Theoretic Topology, Amsterdam: North-Holland, pp. 111–167, 1984.

R. Engelking, General Topology, Revised and completed edition, Berlin : Heldermann Verlag, 1989.

W. M. Fleischman, “A new extension of countable compactness”, Fund. Math., vol. 67, no. 1, pp. 1–9, 1971.

L. Gillman and M. Jerison, Rings of Continuous Functions, New York: Van Nostrand, 1960.

W. Just, A. W. Miller, M. Scheepers and P. J. Szeptycki, “The combinatorics of open covers II”. Topology Appl.. vol. 73, pp. 241-266, 1996.

D. Kocev, “Menger-type covering properties of topological spaces”, Filomat, vol. 29, no. 1, pp. 99–106, 2015.

D. Kocev, “Almost Menger and related spaces”, Mat. Vesnik, vol. 61, no. 2, pp. 173–180, 2009.

Lj. D. R. Kočinac, “Star-Menger and related spaces”, Publ. Math. Debrecen, vol. 55, no. 3-4, pp. 421–431, 1999.

Lj. D. R. Kočinac, “Star-Menger and related spaces II”, Filomat, no. 13, pp. 129–140, 1999.

Lj. D. R. Kočinac, “Star selection principles: a survey”, Khayyam J. Math., vol. 1, no.1, pp. 82-106, 2015.

Lj. D. R. Kočinac, “Variations of classical selection principles: An overview”, Quaest. Math., vol. 43 (2020), no.8, pp. 1121-1153, 2020.

Lj. D. R Kočinac and C. Guido, “Relative covering properties”, Questions Answers Gen. Topology, vol. 19, no. 1, pp. 107-114, 2001.

M. V. Matveev, “Absolutely countably compact spaces”, Topology Appl., vol. 58, no.1, pp. 81–92, 1994.

M. V. Matveev, “Properties close to pseudocompactness and countable compactness”, Vestnik Moskov. Ser. I Mat. Mekh., no. 2, pp. 24-27, 1984.

M. V. Matveev, “A survey on star covering properties”, Topology Atlas (1998), Preprint No. 330.

B. A. Pansera, “Weaker forms of the Menger property”, Quaest. Math., vol. 35, no. 2, pp. 161-169, 2013.

M. Scheepers, “Combinatorics of open covers (I): Ramsey theory”, Topology Appl., vol. 69, no. 1, pp. 31-62, 1992.

Y.-K. Song, “Remarks on strongly star-Menger spaces”, Comment. Math. Univ. Carolin., vol. 54, no. 1, pp. 97–104, 2013.

Y.-K. Song, “On countable star-covering properties”, Appl. Gen. Topol., vol. 8, no. 2, pp. 249–258, 2007.

Y.-K. Song, “Absolutely strongly star-Menger spaces”, Topology Appl., vol 160, no. 3, pp. 475–481, 2013.

Y.-K. Song, “Some remarks on almost star countable spaces”, Studia Sci. Math. Hungar., vol. 52, no. 2, pp. 12–20, 2015.

L. A. Steen and J. A. Seebach, Counterexamples in Topology, New York: Dover Publications, 1995.

R. C. Walker, The Stone-Čech Compactification, Ergebnisse der Mathematik und ihrer Gren- zgebiete, Band 83, New York-Berlin: Springer, 1974.

How to Cite
G. Kumar and B. K. Tyagi, “Weakly strongly star-Menger spaces”, CUBO, vol. 23, no. 2, pp. 287–298, Aug. 2021.