More on Cardinality Bounds Involving the Weak Lindelöf degree

Abstract

We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindelöf degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\le 2^{wL(X)\ddot{I}€\ddot{I}‡(X)\ddot{I}ˆ(X)}$, and if $X$ is additionally power homogeneous, then $|X|\le 2^{wL(X)\ddot{I}€\ddot{I}‡(X)}$. We also prove that if $X$ is an almost Lindelöf space with a strong $G_{\hat{I}}´$-diagonal of rank 2, then $|X|\le 2^{{\aleph }_0}$; that if $X$ is a star-cdc space with a $G_{\hat{I}}´$-diagonal of rank 3, then $|X|\le 2^{{\aleph }_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_{\hat{I}}´$-diagonal of rank 2, then $|X|\le 2^{{\aleph }_0}$. Several improvements of results in [9] are also given. We show that if $X$ is locally compact, then $|X|\le wL(X{)}^{\ddot{I}ˆ(X)}$ and that $|X|\le wL(X{)}^{t(X)}$ if $X$ is additionally power homogeneous. We also prove that $|X|\le 2^{\ddot{I}{ˆ}_c(X)t(X)wL(X)}$ for any space with a $\ddot{I}€$-base whose elements have compact closures and that the stronger inequality $|X|\le wL(X{)}^{\ddot{I}{ˆ}_c(X)t(X)}$ is true when $X$ is locally $H$-closed or locally Lindelöf.

15 pages