Mathematica Bohemica, Vol. 141, No. 4, pp. 457-461, 2016
Cardinalities of DCCC normal spaces with a rank 2-diagonal
Wei-Feng Xuan, Wei-Xue Shi
Received June 6, 2015. First published August 8, 2016.
Abstract: A topological space $X$ has a rank 2-diagonal if there exists a diagonal sequence on $X$ of rank $2$, that is, there is a countable family $\{\mathcal U_n n\in\omega\}$ of open covers of $X$ such that for each $x \in X$, $\{x\}=\bigcap\{ St^2(x, \mathcal U_n) n \in\omega\}$. We say that a space $X$ satisfies the Discrete Countable Chain Condition (DCCC for short) if every discrete family of nonempty open subsets of $X$ is countable. We mainly prove that if $X$ is a DCCC normal space with a rank 2-diagonal, then the cardinality of $X$ is at most $\mathfrak c$. Moreover, we prove that if $X$ is a first countable DCCC normal space and has a $G_\delta$-diagonal, then the cardinality of $X$ is at most $\mathfrak c$.
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Affiliations: Wei-Feng Xuan, College of Science, Nanjing Audit University, 86 YuShan West Road, Nanjing, China, 211815, e-mail: wfxuan@nau.edu.cn; Wei-Xue Shi, Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing, China, 210093, e-mail: wxshi@nju.edu.cn
