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

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 $|X|\leq 2^{wL(X)\pi\chi(X)\psi(X)}$,andif$X$isadditionallypowerhomogeneous,then$|X|\leq 2^{wL(X)\pi\chi(X)}$\mathrm{.}Wealsoprovethatif$X$ is an almost Lindel"of space with a strong $G_{\delta }$-diagonal of rank 2, then $|X|\leq 2^{\aleph_0}$;thatif$X$isastar-cdcspacewitha$G_\delta$-diagonal of rank 3, then $\mathrm{\mid }X\mathrm{\mid }\le 2^{{\mathrm{\aleph }}_0}$; and if $X$ is any normal star-cdc space $X$ with a $G_{\delta }$-diagonal of rank 2, then $\mathrm{\mid }X\mathrm{\mid }\le 2^{{\mathrm{\aleph }}_0}$. Several improvements of results in [9] are also given. We show that if $X$ is locally compact, then $\mathrm{\mid }X\mathrm{\mid }\le wL(X{)}^{\psi (X)}$ and that $\mathrm{\mid }X\mathrm{\mid }\le wL(X{)}^{t(X)}$ if $X$ is additionally power homogeneous. We also prove that $|X|\leq 2^{\psi_c(X)t(X)wL(X)}$foranyspacewitha$\pi$-base whose elements have compact closures and that the stronger inequality $|X|\leq wL(X)^{\psi_c(X)t(X)}$istruewhen$X$islocally$H$-closed or locally Lindel"of.