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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 $\mathfrak c$.

Keywords: cardinality; Discrete Countable Chain Condition; normal space; rank 2-diagonal; $G_\delta$-diagonal

Classification (MSC 2010): 54D20, 54E35

DOI: 10.21136/MB.2016.0027-15

Full text available as PDF.

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