Paper detail

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

For a Urysohn space $X$ we define the regular diagonal degree $\overlineΔ(X)$ of $X$ to be the minimal infinite cardinal $κ$ such that $X$ has a regular $G_κ$-diagonal i.e. there is a family $(U_η:η<κ)$ of open neighborhoods of $Δ_X=\{(x,x)\in X^2:x\in X\}$ in $X^2$ such that $Δ_X = \bigcap_{η<κ} \overline{U}_η$. In this paper we show that if $X$ is a Urysohn space then: (1) $|X|\leq 2^{c(X)\cdot\overlineΔ(X)}$; (2) $|X|\leq 2^{\overlineΔ(X)\cdot 2^{wL(X)}}$; (3) $|X|\le wL(X)^{\overlineΔ(X)\cdotχ(X)}$; and (4) $|X|\le aL(X)^{\overlineΔ(X)}$; where $χ(X)$, $c(X)$, $wL(X)$ 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 $G_δ$-diagonal does not exceed $2^ω$. It follows from (2) that every weakly Lindelöf space with a regular $G_δ$-diagonal has cardinality at most $2^{2^ω}$. Inequality (3) implies that when $X$ is a space with a regular $G_δ$-diagonal then $|X|\le wL(X)^{χ(X)}$. This improves significantly Bell, Ginsburg and Woods inequality $|X|\le 2^{χ(X)wL(X)}$ for the class of normal spaces with regular $G_δ$-diagonals. In particular (3) shows that the cardinality of every first countable space with a regular $G_δ$-diagonal does not exceed $wL(X)^ω$. For the class of spaces with regular $G_δ$-diagonals (4) improves Bella and Cammaroto inequality $|X|\le 2^{χ(X)\cdot aL(X)}$, which is valid for all Urysohn spaces. Also, it follows from (4) that the cardinality of every space with a regular $G_δ$-diagonal does not exceed $aL(X)^ω$.