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Keywords:
star-compact spaces; spaces star determined by a finite number of convergent sequences; $(a)$-spaces; selectively $(a)$-spaces
Summary:
We prove a number of results on star covering properties which may be regarded as either generalizations or specializations of topological properties related to the ones mentioned in the title of the paper. For instance, we give a new, entirely combinatorial proof of the fact that $\Psi$-spaces constructed from infinite almost disjoint families are not star-compact. By going a little further we conclude that if $X$ is a star-compact space within a certain class, then $X$ is neither first countable nor separable. We also show that if a topological space is pseudonormal and has countable extent, then its Alexandroff duplicate satisfies property $(a)$. A number of problems and questions are also presented.
References:
[1] Bella A., Bonanzinga M., Matveev M.: Variations of selective separability. Topology Appl. 156 (2009), no. 7, 1241–1252. DOI 10.1016/j.topol.2008.12.029 | MR 2502000 | Zbl 1197.54038
[2] Bella A., Matveev M., Spadaro S.: Variations of selective separability II: Discrete sets and the influence of convergence and maximality. Topology Appl. 159 (2012), no. 1, 253–271. DOI 10.1016/j.topol.2011.09.005 | MR 2852970 | Zbl 1239.54014
[3] Caserta A., Di Maio G., Kočinac Lj.D.R.: Versions of properties $(a)$ and $(pp)$. Topology Appl. 158 (2011), no. 12, 1360–1368. MR 2812488 | Zbl 1229.54029
[4] Comfort W.W.: Cofinal families in certain function spaces. Commen. Math. Univ. Carolin. 29 (1988), no. 4, 665–675. MR 0982784 | Zbl 0666.54002
[5] van Douwen E.K.: The integers and topology. in K. Kunen, J. Vaughan (Eds.), Handbook of Set-theoretic Topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 0776622 | Zbl 0561.54004
[6] van Douwen E.K., Reed G.M., Roscoe A.W., Tree I.J.: Star covering properties. Topology Appl. 39 (1991), no. 1, 71–103. DOI 10.1016/0166-8641(91)90077-Y | MR 1103993 | Zbl 0743.54007
[7] Engelking R.: General Topology. second edition, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[8] Hodel R.: Cardinal functions, I. in Handbook of Set Theoretic Topology (eds. K. Kunen and J.E. Vaughan), North Holland, Amsterdam, 1984, pp. 1–61. MR 0776620 | Zbl 0559.54003
[9] Just W., Matveev M.V., Szeptycki P.J.: Some results on property $(a)$. Topology Appl. 100 (2000), no. 1, 67–83. MR 1731705 | Zbl 0944.54014
[10] Kočinac Lj.D.R.: Star selection principles: a survey. Khayyam J. Math. 1 (2015), no. 1, 82–106. MR 3353479 | Zbl 1337.54001
[11] Matveev M.V.: Absolutely countably compact spaces. Topology Appl. 58 (1994), no. 1, 81–92. DOI 10.1016/0166-8641(94)90074-4 | MR 1280711 | Zbl 0801.54021
[12] Matveev M.V.: Some questions on property $(a)$. Questions Answers Gen. Topology 15 (1997), no. 2, 103–111. MR 1472172 | Zbl 1002.54016
[13] Matveev M.V.: A survey on star covering properties. Topology Atlas, Preprint 330, 1998.
[14] van Mill J., Tkachuk V.V., Wilson R.G.: Classes defined by stars and neighbourhood assignments. Topology Appl. 154 (2007), no. 10, 2127–2134. DOI 10.1016/j.topol.2006.03.029 | MR 2324924 | Zbl 1131.54022
[15] Morgan C.J.G., da Silva S.G.: Selectively $(a)$-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle. Houston J. Math. 42 (2016), no. 3, 1031–1046. MR 3570723 | Zbl 1371.54142
[16] Mrówka S.: On completely regular spaces. Fund. Math. 41 (1954), 105–106. DOI 10.4064/fm-41-1-105-106 | MR 0063650 | Zbl 0055.41304
[17] Santana H.L.: Classes de espaços definidas por estrelas e atribuiçoes de vizinhanças abertas. MsC Dissertation (in Portuguese), UFBA, 2014, 89 pp.
[18] da Silva S.G.: $(a)$-spaces and selectively $(a)$-spaces from almost disjoint families. Acta Math. Hungar. 142 (2014), no. 2, 420–432. DOI 10.1007/s10474-013-0372-2 | MR 3165490 | Zbl 1299.54046
[19] da Silva S.G.: On the extent of separable, locally compact, selectively $(a)$-spaces. Colloq. Math. 141 (2015), no. 2, 199–208. DOI 10.4064/cm141-2-5 | MR 3404263 | Zbl 1344.54003
[20] Song Y.: Remarks on cs-starcompact spaces. Commun. Korean Math. Soc. 27 (2012), no. 1, 201–205. DOI 10.4134/CKMS.2012.27.1.201 | MR 2919026 | Zbl 1234.54035
[21] Song Y.: Remarks on selectively $(a)$-spaces. Topology Appl. 160 (2013), no. 6, 806–811. DOI 10.1016/j.topol.2013.02.003 | MR 3028613 | Zbl 1262.54006
[22] Szeptycki P.J., Vaughan J.E.: Almost disjoint families and property $(a)$. Fund. Math. 158 (1998), no. 3, 229–240. MR 1663330 | Zbl 0933.54005
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