Geodesic
Cardinalities of DCCC normal spaces with a rank 2-diagonal
Mathematica Bohemica, Tome 141 (2016) no. 4, pp. 457-461
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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 $\{\mathcal U_n\colon n\in \omega \}$ of open covers of $X$ such that for each $x \in X$, $\{x\}=\bigcap \{{\rm St}^2(x, \mathcal U_n)\colon n \in \omega \}$. 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 $\mathfrak c$.
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 $\{\mathcal U_n\colon n\in \omega \}$ of open covers of $X$ such that for each $x \in X$, $\{x\}=\bigcap \{{\rm St}^2(x, \mathcal U_n)\colon n \in \omega \}$. 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 $\mathfrak c$.
DOI : 10.21136/MB.2016.0027-15
Classification : 54D20, 54E35
Keywords: cardinality; Discrete Countable Chain Condition; normal space; rank 2-diagonal; $G_\delta $-diagonal 
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Xuan, Wei-Feng; Shi, Wei-Xue. Cardinalities of DCCC normal spaces with a rank 2-diagonal. Mathematica Bohemica, Tome 141 (2016) no. 4, pp. 457-461. doi: 10.21136/MB.2016.0027-15

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