Cardinalities of weakly Lindelöf spaces with regular $G_κ$-diagonals

Ivan S. Gotchev

Published 2015-04-08Version 1

For a Urysohn space $X$ we define the \emph{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 sets in $X^2$ such that $\{(x,x)\in X^2:x\in X\} = \cap_{\eta<\kappa} \overline{U}_\eta$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq 2^{wL(X)\cdot\overline{\Delta}(X)}$; (2) $|X|\le wL(X)^{\overline{\Delta}(X)\cdot\chi(X)}$; and (3) $|X|\le aL(X)^{\overline{\Delta}(X)}$. It follows from (1) that every weakly Lindel\"of space with a regular $G_\delta$-diagonal has cardinality at most $2^\omega$. This generalizes Buzyakova's result that the cardinality of a ccc-space with a regular $G_\delta$-diagonal does not exceed $2^\omega$. It also follows that $|X|\leq 2^{\chi(X)\cdot wL(X)\cdot\overline{\Delta}(X)}$; hence Bell, Ginsburg and Woods inequality $|X|\le 2^{\chi(X)wL(X)}$, which is known to be true for normal $T_1$-spaces, is true also for a large class of Urysohn spaces that includes all spaces with regular $G_\delta$-diagonals. Inequality (2) 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 for the class of normal spaces with regular $G_\delta$-diagonals. In particular (2) 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 (3) improves Bella and Cammaroto inequality $|X|\le 2^{\chi(X)\cdot aL(X)}$, which is valid for all Urysohn spaces. Also, it follows from (3) that the cardinality of every space with a regular $G_\delta$-diagonal does not exceed $aL(X)^\omega$.