  Using weaker versions of the cardinal function $\psi_c(X)$, we derive a
series of new bounds for the cardinality of Hausdorff spaces and regular spaces
that do not involve $\psi_c(X)$ nor its variants at all. For example, we show
if $X$ is regular then $|X|\leq 2^{c(X)^{\pi\chi(X)}}$ and $|X|\leq
2^{c(X)\pi\chi(X)^{ot(X)}}$, where the cardinal function $ot(X)$, introduced by
Tkachenko, has the property $ot(X)\leq\min\{t(X),c(X)\}$. It follows from the
latter that a regular space with cellularity at most $\mathfrak{c}$ and
countable $\pi$-character has cardinality at most $2^\mathfrak{c}$. For a
Hausdorff space $X$ we show $|X|\leq 2^{d(X)^{\pi\chi(X)}}$, $|X|\leq
d(X)^{\pi\chi(X)^{ot(X)}}$, and $|X|\leq 2^{\pi w(X)^{dot(X)}}$, where
$dot(X)\leq\min\{ot(X),\pi\chi(X)\}$. None of these bounds involve $\psi_c(X)$
or $\psi(X)$. By introducing the cardinal functions $w\psi_c(X)$ and
$d\psi_c(X)$ with the property $w\psi_c(X)d\psi_c(X)\leq\psi_c(X)$ for a
Hausdorff space $X$, we show $|X|\leq\pi\chi(X)^{c(X)w\psi_c(X)}$ if $X$ is
regular and $|X|\leq\pi\chi(X)^{c(X)d\psi_c(X)w\psi_c(X)}$ if $X$ is Hausdorff.
This improves results of Sapirovskii and Sun. It is also shown that if $X$ is
Hausdorff then $|X|\leq 2^{d(X)w\psi_c(X)}$, which appears to be new even in
the case where $w\psi_c(X)$ is replaced with $\psi_c(X)$. Compact examples show
that $\psi(X)$ cannot be replaced with $d\psi_c(X)w\psi_c(X)$ in the bound
$2^{\psi(X)}$ for the cardinality of a compact Hausdorff space $X$. Likewise,
$\psi(X)$ cannot be replaced with $d\psi_c(X)w\psi_c(X)$ in the
Arhangel'skii-Sapirovskii bound $2^{L(X)t(X)\psi(X)}$ for the cardinality of a
Hausdorff space $X$. Finally, we make several observations concerning
homogeneous spaces in this connection.
