Cardinalities of weakly Lindelöf spaces with regular $G_{\kappa }$-diagonals

Abstract

For a Urysohn space $X$ we define the regular diagonal degree $\overline{\Delta }(X)$ of $X$ to be the minimal infinite cardinal $\kappa$ such that $X$ has a regular $G_{\kappa }$-diagonal i.e. there is a family $(U_{\eta }:\eta <\kappa )$ of open neighborhoods of ${\Delta }_X=\{(x,x)\in X^2:x\in X\}$ in $X^2$ such that ${\Delta }_X={\bigcap }_{\eta <\kappa }{\overline{U}}_{\eta }$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\le 2^{c(X)\cdot \overline{\Delta }(X)}$; (2) $|X|\le 2^{\overline{\Delta }(X)\cdot 2^{wL(X)}}$; (3) $|X|\le wL(X{)}^{\overline{\Delta }(X)\cdot \chi (X)}$; and (4) $|X|\le aL(X{)}^{\overline{\Delta }(X)}$; where $\chi (X)$, $c(X)$, $wL(X)$ and $aL(X)$ are respectively the character, the cellularity, the weak Lindelöf number and the almost Lindelöf number of $X$. The first inequality extends to the uncountable case Buzyakova's result that the cardinality of a ccc-space with a regular $G_{\delta }$-diagonal does not exceed $2^{\omega }$. It follows from (2) that every weakly Lindelöf space with a regular $G_{\delta }$-diagonal has cardinality at most $2^{2^{\omega }}$. Inequality (3) implies that when $X$ is a space with a regular $G_{\delta }$-diagonal then $|X|\le wL(X{)}^{\chi (X)}$. This improves significantly Bell, Ginsburg and Woods inequality $|X|\le 2^{\chi (X)wL(X)}$ for the class of normal spaces with regular $G_{\delta }$-diagonals. In particular (3) shows that the cardinality of every first countable space with a regular $G_{\delta }$-diagonal does not exceed $wL(X{)}^{\omega }$. For the class of spaces with regular $G_{\delta }$-diagonals (4) improves Bella and Cammaroto inequality $|X|\le 2^{\chi (X)\cdot aL(X)}$, which is valid for all Urysohn spaces. Also, it follows from (4) that the cardinality of every space with a regular $G_{\delta }$-diagonal does not exceed $aL(X{)}^{\omega }$.

12 pages