On localization of the star-Menger selection principle

Abstract

In this paper we primarily introduce the local version of star-Menger property, namely locally star-Menger property (a space with this property is called locally star-Menger) and present some important topological observations. Certain interactions between the new notion and star-Menger property are also observed. Some observations on effectively locally star-Menger Pixley-Roy hyperspaces (introduced here) are obtained. Preservation like properties under several topological operations are also interpreted carefully. Besides, several results on decomposition and remainder of locally star-Menger spaces are also presented.

Keywords

• locally star-Menger
• effectively locally star-Menger
• Pixley-Roy hyperspace
• decomposition
• remainder
• star-Menger
• Menger

References

1. [1] A.V. Arhangel’skii, A class of spaces which contains all metric and all locally compact spaces, Mat. Sb. (N.S.), 67 (109), 55–88, 1965, English translation: Amer. Math. Soc. Transl. 92, 1–39, 1970.
2. [2] A.V. Arhangel’skii, Mappings and spaces, Russian Math. Surveys, 21 (4), 115–162, 1966.
3. [3] A.V. Arhangel’skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150, 79–90, 2005.
4. [4] A.V. Arhangel’skii, A generalization of Čech-complete spaces and Lindelöf $\Sigma$-spaces, Comment. Math. Univ. Carolin. 54 (2), 121–139, 2013.
5. [5] A.V. Arhangel’skii, Remainders of metrizable and close to metrizable spaces, Fund. Math. 220, 71–81, 2013.
6. [6] A.V. Arhangel’skii and M.M. Choban, Some generalizations of the concept of a $p$-space, Topology Appl. 158, 1381–1389, 2011.
7. [7] A.S. Besicovitch, Concentrated and rarified sets of points, Acta Math. 62, 289–300, 1934.
8. [8] M. Bonanzinga and M. Matveev, Some covering properties for $\Psi$-spaces, Mat. Vesnik, 61, 3–11, 2009.

9. [9] J. Casas-de la Rose, S.A. Garcia-Balan and P.J. Szeptycki, Some star and strongly star selection principles, Topology Appl. 258, 572–587, 2019.
10. [10] J. Cruz-Castillo, A. Ramírez-Páramo and J.F. Tenorio, Menger and Menger-type star selection principles for hit-and-miss topology, Topology Appl. 290, Art. ID 107574, 2021.
11. [11] P. Daniels, Pixley-Roy spaces over subsets of the reals, Topology Appl. 29, 93–106, 1988.
12. [12] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.
13. [13] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, NJ, 1960.
14. [14] K.P. Hart, J. Nagata and J.E. Vaughan, Encyclopedia of General Topology, Elsevier Science Publishers B. V., Amsterdam, 2004.
15. [15] W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, Math. Z. 24, 401–421, 1926.
16. [16] W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, The combinatorics of open covers (II), Topology Appl. 73, 241–266, 1996.
17. [17] Lj.D.R. Kočinac, Star-Menger and related spaces, Publ. Math. Debrecen, 55, 421–431, 1999.
18. [18] Lj.D.R. Kočinac, The Pixley-Roy topology and selection principles, Questions Answers Gen. Topology, 19 (2), 219–225, 2001.
19. [19] Lj.D.R. Kočinac, Star selection principles: A survey, Khayyam J. Math. 1 (1), 82– 106, 2015.
20. [20] K. Menger, Einige Überdeckungssätze de Punktmengenlehre, Sitzungsber. Wien. Abt. 2a Math. Astronom. Phys. Meteorol. Mech. 133, 421–444, 1924.
21. [21] E.A. Michael, Bi-quotient maps and cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble), 18 (2), 287–302, 1968.
22. [22] S. Mrówka, On completely regular spaces, Fund. Math. 41, 105–106, 1954.
23. [23] K. Nagami, $\Sigma$-spaces, Fund. Math. 65, 169–192, 1969.
24. [24] C. Pixley and P. Roy, Uncompletable Moore spaces, in: Proceedings Auburn University Topology Conference, 75–85, 1969.
25. [25] M. Sakai, Star versions of the Menger property, Topology Appl. 176, 22–34, 2014.
26. [26] M. Scheepers, Combinatorics of open covers I: Ramsey theory, Topology Appl. 69, 31–62, 1996.
27. [27] Y.-K. Song, On star-K-Menger spaces, Hacet. J. Math. Stat. 43 (5), 769–776, 2014.
28. [28] L.A. Steen and J.A. Seebach Jr., Counterexamples in Topology, Springer, New York, 1978.
29. [29] P. Szewczak and B. Tsaban, Products of Menger spaces: A combinatorial approach, Ann. Pure Appl. Logic, 168 (1), 1–18, 2017.
30. [30] B. Tsaban, Combinatorial aspects of selective star covering properties in $\Psi$-spaces, Topology Appl. 192, 198–207, 2015.
31. [31] H. Wang and W. He, On remainders of locally s-spaces, Topology Appl. 278, Art. ID 107231, 2020.
32. [32] G. Whyburn and E. Duda, Dynamic Topology, Springer, New York, 1979.
33. [33] S. Willard, General Topology, Addison Wesley Publishing Co., 1970.