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march 2018 Observations on spaces with property $(DC(\omega_1))$
Wei-Feng Xuan, Wei-Xue Shi
Bull. Belg. Math. Soc. Simon Stevin 25(1): 55-62 (march 2018). DOI: 10.36045/bbms/1523412052

Abstract

A topological space $X$ has property $(DC(\omega_1))$ if it has a dense subspace every uncountable subset of which has a limit point in $X$. In this paper, we make some observations on spaces with property $(DC(\omega_1))$. In particular, we prove that the cardinality of a space $X$ with property $(DC(\omega_1))$ does not exceed $\mathfrak c$ if $X$ satisfies one of the following conditions: (1) $X$ is normal and has a rank $2$-diagonal; (2) $X$ is perfect and has a rank $2$-diagonal; (3) $X$ has a rank $3$-diagonal; (4) $X$ is perfect and has countable tightness. We also prove that if $X$ is a regular space with a $G_\delta$-diagonal and property $(DC(\omega_1))$ then the cardinality of $X$ is at most $2^\mathfrak c$.

Citation

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Wei-Feng Xuan. Wei-Xue Shi. "Observations on spaces with property $(DC(\omega_1))$." Bull. Belg. Math. Soc. Simon Stevin 25 (1) 55 - 62, march 2018. https://doi.org/10.36045/bbms/1523412052

Information

Published: march 2018
First available in Project Euclid: 11 April 2018

zbMATH: 06882541
MathSciNet: MR3784505
Digital Object Identifier: 10.36045/bbms/1523412052

Subjects:
Primary: 54D20
Secondary: 54E35

Rights: Copyright © 2018 The Belgian Mathematical Society

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Vol.25 • No. 1 • march 2018
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