Cardinalities of weakly Lindel\"of 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) $|X|\leq 2^{c(X)\cdot\overline{\Delta}(X)}$; (2) $|X|\leq 2^{\overline{\Delta}(X)\cdot 2^{wL(X)}}$; (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 Lindel\"of number and the almost Lindel\"of 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 Lindel\"of 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 $aL(X)^\omega$.