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A note on resolvability

  • K. P. S. Bhaskara RaoEmail author
Article
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Abstract

A topological space is said to be resolvable if it has two disjoint dense subsets. If \({\aleph}\) is a cardinal number(finite or infinite), a topological space is said to be \({\aleph}\) -resolvable if it has a paiwise disjoint family of \({\aleph}\) many dense subsets. Illanes [3] showed that a topological space which is \({\kappa}\) -resolvable for every finite integer \({\kappa}\) is necessarily \({\aleph_{0}}\) -resolvable. We generalize this result to infinite cardinals. We show that if a topological space X is \({\kappa}\) -resolvable for every \({\kappa}\) < \({\aleph}\) and if cofinality of \({\aleph}\) is \({\aleph_{0}}\) , then, X is \({\aleph}\)-resolvable.

Key words and phrases

\({\aleph}\)-resolvability \({\aleph}\)-\({\ast}\)-resolvability 

Mathematics Subject Classification

54A10 54A35 54D05 54D10 54G20 54E52 

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References

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    Comfort W.W., Feng Li.: The union of resolvable spaces is resolvable. Math. Japon., 38, 413–414 (1993)MathSciNetzbMATHGoogle Scholar
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    Comfort W.W., Garcia-Ferreira Salvador., Resolvability: a selective survey and some new results. Topology Appl., 74, 149–167 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
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    Illanes Alejandro.: Finite and \({\omega}\) -resolvability. Proc. Amer. Math. Soc., 124, 1243–1246 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
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    Oleg Pavlov, Problems on (ir)resolvability, in: Open Problems in Topology, II (edited by E. M. Pearl), Elsevier (Amsterdam, 2007), pp. 51–59.Google Scholar
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    Zoltán Szentmiklóssy, Resolvability of topological spaces, Thesis submitted to Central European University (Budapest, 2009).Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Department of Computer Information SystemsIndiana University NorthwestGaryUSA

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