Abstract
Hajnal and Juhász [9] proved that if X is a T1-space, then \({|X| \leq 2^{s(X)\psi(X)}}\), and if X is a Hausdorff space, then \({|X| \leq 2^{c(X)\chi(X)}}\) and \({|X| \leq 2^{2^{s(X)}}}\). Schröder sharpened the first two estimations by showing that if X is a Hausdorff space, then \({|X| \leq 2^{Us(X)\psi_c(X)}}\), and if X is a Urysohn space, then \({|X| \leq 2^{Uc(X)\chi(X)}}\).
In this paper, for any positive integer n and some topological spaces X, we define the cardinal functions \({\chi_n(X), \psi_n(X), s_n(X)}\), and cn(X) called respectively S(n)-character, S(n)-pseudocharacter, S(n)-spread, and S(n)-cellularity and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of S(n)-spaces. We recall that the S(1)-spaces are exactly the Hausdorff spaces and the S(2)-spaces are exactly the Urysohn spaces.
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References
O. T. Alas and Lj. D. R. Kočinac, More cardinal inequalities on Urysohn spaces, Math. Balkanica (N.S.), 14 (2000), 247–251
Charlesworth, A.: On the cardinality of a topological space. Proc. Amer. Math. Soc. 66, 138–142 (1977)
Dikranjan, D., Giuli, E.: \(S(n)\)-\(\theta \)-closed spaces. Topology Appl. 28, 59–74 (1988)
D. Dikranjan and W. Tholen, Categorical Structure of Closure Operators, with Applications to Topology, Algebra and Discrete Mathematics, Mathematics and its Applications, 346, Kluwer Academic Publishers Group (Dordrecht, 1995)
Dikranjan, D., Watson, S.: The category of \(S(\alpha )\)-spaces is not cowellpowered. Topology Appl. 61, 137–150 (1995)
R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, revised ed., Heldermann Verlag (Berlin, 1989)
Gotchev, I.S.: Cardinal inequalities for Urysohn spaces involving variations of the almost Lindelöf degree. Serdica Math. J. 44, 195–212 (2018)
I. S. Gotchev and Lj. D. R. Kočinac, More on the cardinality of \(S(n)\)-spaces, Serdica Math. J., 44 (2018), 227 – 242
A. Hajnal and I. Juhász, Discrete subspaces of topological spaces, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math., 29 (1967), 343–356
R. E. Hodel, Cardinal functions. I, in: Handbook of Set-Theoretic Topology, (K. Kunen and J. E. Vaughan, eds.) North-Holland (Amsterdam, 1984), pp. 1–61
I. Juhász, Cardinal Functions in Topology—Ten Years Later, Mathematical Centre Tracts, No. 123, Mathematisch Centrum (Amsterdam, 1980)
Pol, R.: Short proofs of two theorems on cardinality of topological spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22, 1245–1249 (1974)
Porter, J.R., Votaw, C.: \(S(\alpha )\) spaces and regular Hausdorff extensions. Pacific J. Math. 45, 327–345 (1973)
Schröder, J.: Urysohn cellularity and Urysohn spread. Math. Japon. 38, 1129–1133 (1993)
N. V. Veličko, \(H\)-closed topological spaces, Mat. Sb. (N.S.), 70 (1966), 98–112
Viglino, G.: \(\bar{T}_{n}\)-spaces. Kyungpook Math. J. 11, 33–35 (1971)
Acknowledgement
The author is grateful to the anonymous referee for very careful reading of the paper and for several valuable suggestions.
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Some of the results in this paper were announced at the Spring Topology and Dynamical Systems Conference, Berry College, Mount Berry, GA, March 17–19, 2005.
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Gotchev, I.S. Cardinal inequalities for S(n)-spaces. Acta Math. Hungar. 159, 229–245 (2019). https://doi.org/10.1007/s10474-019-00939-0
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DOI: https://doi.org/10.1007/s10474-019-00939-0
Key words and phrases
- cardinal function
- S(n)-space
- S(n)-character
- S(n)-pseudocharacter
- S(n)-discrete
- S(n)-spread
- S(n)-cellularity