  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.

General Topology