More on the cardinality of a topological space
Keywords:n-Hausdorff space, n-Urysohn space, homogeneous spaces, cardinal invariants
In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel’skii, Alas, Hajnal-Juhász, Bell-Gisburg-Woods, Dissanayake-Willard, Schröder and to the excellent survey by Hodel “Arhangel’skii’s Solution to Alexandroff’s problem: A survey”.
Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel’skii’s question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindelöf space having countable pseudo-character (i.e., points are Gδ). Shelah in 1978 was the first to give a consistent negative answer to Arhangel’skii’s question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995. The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindelöf spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight Hpw(X), and prove that (1) |X| ≤ Hpsw(X)aLc(X)ψ(X), for Hausdorff spaces and (2) |X| ≤ Hpsw(X)ωLc(X)ψ(X), where X is a Hausdorff space with a π-base consisting of compact sets with non-empty interior. In 1993 Schröder proved an analogue of Hajnal and Juhasz inequality |X| ≤ 2c(X)χ(X) for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity Uc(X) instead of cellularity c(X). We introduce the n-Urysohn cellularity n-Uc(X) (where n≥2) and prove that the previous inequality is true in the class of n-Urysohn spaces replacing Uc(X) by the weaker n-Uc(X). We also show that |X| ≤ 2Uc(X)πχ(X) if X is a power homogeneous Urysohn space.
O. T. Alas, More topological cardinal inequalities, Colloq. Math. 65, no. 2 (1993), 165-168. https://doi.org/10.4064/cm-65-2-165-168
A. Arhangel'skii, On cardinal invariants, General Topology and Its Relations to Modern Analysis and Algebra IV, Academic Press, New York, 1972, 37-46.
A. V. Arhangel'skii, The power of bicompacta with first axiom of countability, Sov. Math. Dokl. 10 (1969), 951-955.
A. V. Arhangel'skii, A theorem about cardinality, Russian Math. Surveys 34 (1979), 303-325. https://doi.org/10.1070/RM1979v034n04ABEH002968
M. Bell, J. Gisburg and G. Woods, Cardinal inequalities for topological spaces involving the weak Lindelöf number, Pacific J. Math. 79, no. 1 (1978), 37-45. https://doi.org/10.2140/pjm.1978.79.37
M. Bonanzinga, On the Hausdorff number of a topological space, Houston J. Math. 39, no. 3 (2013), 1013-1030.
M. Bonanzinga, F. Cammaroto and M. V. Matveev, On the Urysohn number of a topological space, Quaest. Math. 34, no. 4 (2011), 441-446. https://doi.org/10.2989/16073606.2011.640456
N. A. Carlson and G. J. Ridderbos, Partition relations and power homogeneity, Top. Proc. 32 (2008), 115-124.
A. Charlesworth, On the cardinality of a topological space, Proc. Amer. Math. Soc. 66, no. 1 (1977), 138-142. https://doi.org/10.1090/S0002-9939-1977-0451184-8
U. N. B. Dissanayake and S. Willard, The almost Lindelöf degree, Canad. Math. Bull. 27, no. 4 (1984), 452-455. https://doi.org/10.4153/CMB-1984-070-2
I. Gorelic, The Baire category and forcing large Lindelöf spaces with points $G_delta$, Proc. Amer. Math. Soc. 118 (1993), 603-607. https://doi.org/10.2307/2160344
I. Gotchev, Generalizations of two cardinal inequalities of Hajnal and Juhász, arXiv: 1504.01790[math.GN]
A. Hajnal and I. Juhász, Discrete subspaces of topological spaces, Indag. Math. 29 (1967), 343-356. https://doi.org/10.1016/S1385-7258(67)50048-3
R. E. Hodel, Arhangel'skii's solution to Alexandroff's problem: A survey, Topol. Appl. 153, no. 13 (2006), 2199-2217. https://doi.org/10.1016/j.topol.2005.04.011
I. Juhász, Cardinal Functions in Topology - Ten Years Later, Math.l Centre Tracts 123, Amsterdam (1980).
J. Schröder, Urysohn cellularity and Urysohn spread, Math. Japonica 38 (1993), 1129-1133.
S. Shelah, On some problems in general topology, Contemporary Mathematics 192 (1996), 91-101. https://doi.org/10.1090/conm/192/02352
F. Tall, On the cardinality of Lindelöf spaces with points $G_delta$, Topology Appl. 63 (1995), 21-38. https://doi.org/10.1016/0166-8641(95)90002-0
N. V. Velicko, H-closed topological spaces, Mat. Sb. (N.S.) 70 (112) (1996), 98-112 (in Russian).
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