Cardinalities of DCCC normal spaces with a rank 2-diagonal

Xuan, Wei-Feng, and Shi, Wei-Xue. "Cardinalities of DCCC normal spaces with a rank 2-diagonal." Mathematica Bohemica 141.4 (2016): 457-461. <http://eudml.org/doc/287536>.

@article{Xuan2016,
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 $\lbrace \mathcal \{U\}_n\colon n\in \omega \rbrace $ of open covers of $X$ such that for each $x \in X$, $\lbrace x\rbrace =\bigcap \lbrace \{\rm St\}^2(x, \mathcal \{U\}_n)\colon n \in \omega \rbrace $. 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 $\mathfrak \{c\}$.},
author = {Xuan, Wei-Feng, Shi, Wei-Xue},
journal = {Mathematica Bohemica},
keywords = {cardinality; Discrete Countable Chain Condition; normal space; rank 2-diagonal; $G_\delta $-diagonal},
language = {eng},
number = {4},
pages = {457-461},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cardinalities of DCCC normal spaces with a rank 2-diagonal},
url = {http://eudml.org/doc/287536},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Xuan, Wei-Feng
AU - Shi, Wei-Xue
TI - Cardinalities of DCCC normal spaces with a rank 2-diagonal
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 4
SP - 457
EP - 461
AB - 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 $\lbrace \mathcal {U}_n\colon n\in \omega \rbrace $ of open covers of $X$ such that for each $x \in X$, $\lbrace x\rbrace =\bigcap \lbrace {\rm St}^2(x, \mathcal {U}_n)\colon n \in \omega \rbrace $. 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 $\mathfrak {c}$.
LA - eng
KW - cardinality; Discrete Countable Chain Condition; normal space; rank 2-diagonal; $G_\delta $-diagonal
UR - http://eudml.org/doc/287536
ER -