More on Cardinality Bounds Involving the Weak Lindel\"of degree

Abstract

We give several new bounds for the cardinality of a Hausdorff topological space $X$ involving the weak Lindel\"of degree $wL(X)$. In particular, we show that if $X$ is extremally disconnected, then $|X|\leq 2^{wL(X)\pi\chi(X)\psi(X)}$, and if $X$ is additionally power homogeneous, then $|X|\leq 2^{wL(X)\pi\chi(X)}$. We also prove that if $X$ is an almost Lindel\"of space with a strong $G_\delta$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$; that if $X$ is a star-cdc space with a $G_\delta$-diagonal of rank 3, then $|X| \le 2^{\aleph_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_\delta$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$. Several improvements of results in [9] are also given. We show that if $X$ is locally compact, then $|X|\leq wL(X)^{\psi(X)}$ and that $|X|\leq wL(X)^{t(X)}$ if $X$ is additionally power homogeneous. We also prove that $|X|\leq 2^{\psi_c(X)t(X)wL(X)}$ for any space with a $\pi$-base whose elements have compact closures and that the stronger inequality $|X|\leq wL(X)^{\psi_c(X)t(X)}$ is true when $X$ is locally $H$-closed or locally Lindel\"of.