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Keywords:
relative topological property; Lindelöf; star-Lindelöf; relative extent; relative property (a)
Summary:
In this paper, we prove the following statements: (1) For every regular uncountable cardinal $\kappa $, there exist a Tychonoff space $X$ and $Y$ a subspace of $X$ such that $Y$ is both relatively absolute star-Lindelöf and relative property (a) in $X$ and $e(Y,X) \ge \kappa $, but $Y$ is not strongly relative star-Lindelöf in $X$ and $X$ is not star-Lindelöf. (2) There exist a Tychonoff space $X$ and a subspace $Y$ of $X$ such that $Y$ is strongly relative star-Lindelöf in $X$ (hence, relative star-Lindelöf), but $Y$ is not absolutely relative star-Lindelöf in $X$.
References:
[1] A. V.  Arhangel’skii, M. M. Genedi Hamdi: The origin of the theory of relative topological properties. General Topology, Space and Mappings, Moskov. Gos. Univ., Moscow, 1989, pp. 3–48. (Russian) MR 1095304
[2] A. V.  Arhangel’skii: A generic theorem in the theory of cardinal invariants of topological spaces. Comment. Math. Univ. Carolinae 36 (1995), 303–325. MR 1357532
[3] A. V.  Arhangel’skii: Relative topological properties and relative topological spaces. Topology Appl. 70 (1996), 87–99. DOI 10.1016/0166-8641(95)00086-0 | MR 1397067
[4] M.  Bonanzinga: Star-Lindelöf and absolutely star-Lindelöf spaces. Q and A in General Topology 14 (1998), 79–104. MR 1642032 | Zbl 0931.54019
[5] E. K.  van Douwen, G. M.  Reed, A. W.  Roscoe, and I. J.  Tree: Star covering properties. Topology Appl. 39 (1991), 71–103. DOI 10.1016/0166-8641(91)90077-Y | MR 1103993
[6] M. Dai: A class of topological spaces containing Lindelöf spaces and separable spaces. Chin. Ann. Math. Ser.  A 4 (1983), 571–575. MR 0742178
[7] R.  Engelking: General Topology. Rev. and compl. ed. Heldermann-Verlag, Berlin, 1989. MR 1039321 | Zbl 0684.54001
[8] Lj. D.  Kocinac: Some relative topological properties. Mat. Ves. 44 (1992), 33–44. MR 1201265 | Zbl 0795.54002
[9] M. V.  Matveev: Absolutely countably compact spaces. Topology Appl. 58 (1994), 81–92. DOI 10.1016/0166-8641(94)90074-4 | MR 1280711 | Zbl 0801.54021
[10] M. V.  Matveev: A survey on star covering properties. Topology Atlas, preprint No.  330 (1998).
[11] M. V.  Matveev: A survey on star covering properties  II. Topology Atlas, preprint No.  431 (2000).
[12] M. V.  Matveev: Some questions on property  (a). Quest. Answers Gen. Topology 15 (1997), 103–111. MR 1472172 | Zbl 1002.54016
[13] M. V.  Matveev, O. I.  Pavlov, and J. K.  Tartir: On relatively normal spaces, relatively regular spaces, and on relative property  (a). Topology Appl. 93 (1999), 121–129. DOI 10.1016/S0166-8641(97)00265-4 | MR 1680839
[14] M. V.  Matveev: How weak is weak extent?. Topology Appl. 119 (2002), 229–232. DOI 10.1016/S0166-8641(01)00061-X | MR 1886097 | Zbl 0986.54003
[15] M. V.  Matveev: On space in countable web. Preprint.
[16] Y-K.  Song: Spaces with large extent and large star-Lindelöf number. Houston. J.  Math. 29 (2003), 345–352. MR 1987579 | Zbl 1064.54006
[17] Y-K. Song: Discretely star-Lindelöf spaces. Tsukuba J.  Math. 25 (2001), 371–382. MR 1869769 | Zbl 1011.54020
[18] Y-K. Song: On relative star-Lindelöf spaces. N. Z. Math 34 (2005), 159–163. MR 2195833 | Zbl 1099.54022
[19] Y.  Yasui, Z-M. Gao: Spaces in countable web. Houston.  J. Math. 25 (1999), 327–335. MR 1697629
[20] J. E.  Vaughan: Absolute countable compactness and property  (a). Proceedings of the Eighth Prague Topological symposium, August  1996, 1996, pp. 18–24.
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