Abstract
We introduce the class of \(\theta^n\)-Urysohn spaces and the \(n\)-\(\theta\)-closure operator. \(\theta^n\)-Urysohn spaces generalize the notion of a Urysohn space and we consider their relationship with S(n)-spaces, studied in [9], [14] and [18]. We estabilish bounds on the cardinality of these spaces and cardinality bounds if the space is additionally homogeneous.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Basile, F.A., Bonanzinga, M., Carlson, N.: Variations on known and recent cardinality bounds. Topology Appl. 240, 228–237 (2018)
Basile, F.A., Carlson, N.: On the cardinality of Urysohn spaces and weakly H-closed spaces. Math. Bohemica 144, 325–336 (2019)
Bella, A., Cammaroto, F.: On the cardinality of Urysohn spaces. Canad. Math. Bull. 31, 153–158 (1988)
Bonanzinga, M., Carlson, N., Cuzzupé, M.V., Stavrova, D.: More on the cardinality of a topological space. App. Gen. Top. 19, 269–280 (2018)
Cammaroto, F., Catalioto, A., Porter, J.: On the cardinality of Urysohn spaces. Topology Appl. 160, 1862–1869 (2013)
Cammaroto, F., Catalioto, A., Porter, J.: Cardinal functions \(F_{\theta }(X)\) and \(t_{\theta }(X)\) for \(H\)-closed spaces. Quest. Math. 37, 309–320 (2014)
F. Cammaroto and Lj. Kočinac, On \(\theta \)-tightness, Facta Universitatis (Niš), Ser. Math. Inform., 8 (1993), 77–85
Carlson, N., Ridderbos, G.J.: Partition relations and power homogeneity. Topology Proc. 32, 115–124 (2008)
Dikranjan, D., Giuli, E.: \(S(n)\text{-}\theta \)-closed spaces. Topology Appl. 28, 59–74 (1988)
R. Engelking, General Topology, Heldermann Verlag (1989)
Gotchev, I.S.: Cardinal inequalities for Urysohn spaces involving variations of the almost Lindelöf degree. Serdica Math. J. 44, 195–212 (2018)
Hodel, R.E.: Arhangel'skii's solution to Alexandroff's problem: A survey. Topology Appl. 153, 2199–2217 (2006)
A. Osipov, On the cardinality of \(S(n)\)-spaces, arXiv:1809.09587
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. Japonica 38, 1129–1133 (1993)
Shu-Hao, S.: Two new topological cardinal inequality. Proc. Amer. Math. Soc. 104, 313–316 (1988)
N. V. Veličko, H-closed topological spaces, Mat. Sb. (N.S.), 70 (112) (1966), 98–112 (in Russian)
Viglino, G.: \(\overline{T}_{n}\)-spaces. Kyungpook Math. J. 11, 33–35 (1971)
Willard, S., Dissanayake, U.N.B.: The almost Lindelöf degree. Canad. Math. Bull. 27, 452–455 (1984)
Acknowledgements
The authors are very grateful to the anonymous reviewer for valuable comments and suggestions to improve the quality of the paper and for suggesting Question 1. The first author would like to thank also Professor I. Gotchev for good conversations relating \(\theta^n\)-Urysohn spaces and S(n)-spaces.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Basile, F.A., Carlson, N. & Porter, J. On cardinality bounds for \(\theta^n\)-Urysohn spaces. Acta Math. Hungar. 159, 109–123 (2019). https://doi.org/10.1007/s10474-019-00981-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-019-00981-y