More on the cardinality of a topological space

In this paper we continue to investigate the impact that various separation axioms and covering properties have onto the cardinality of topological spaces. Many authors have been working in that field. To mention a few, let us refer to results by Arhangel'skii, Alas, Hajnal-Juh{'a}sz, Bell-Gisburg-Woods, Dissanayake-Willard, Schr"oder and to the excellent survey by Hodel ``Arhangel'ski{\ui}'s Solution to {A}lexandroff's problem: A survey''. Here we provide improvements and analogues of some of the results obtained by the above authors in the settings of more general separation axioms and cardinal invariants related to them. We also provide partial answer to Arhangel'skii's question concerning whether the continuum is an upper bound for the cardinality of a Hausdorff Lindel"of space having countable pseudo-character (i.e., points are $G_delta$). Shelah in 1978 was the first to give a consistent negative answer to Arhangel'skii's question; in 1993 Gorelic established an improved result; and further results were obtained by Tall in 1995. The question of whether or not there is a consistent bound on the cardinality of Hausdorff Lindel"of spaces with countable pseudo-character is still open. In this paper we introduce the Hausdorff point separating weight $Hpw(X)$, and prove that (1) $|X|leq Hpsw(X)^{aL_c(X) psi(X)}$, for Hausdorff spaces and (2) $|X|leq Hpsw(X)^{wL_c(X)psi(X)}$, where $X$ is a Hausdorff space with a $pi$-base consisting of compact sets with non-empty interior. In 1993 Schr"oder proved an analogue of Hajnal and Juhasz inequality $|X|leq 2^{c(X)chi(X)}$ for Hausdorff spaces, for Urysohn spaces by considering weaker invariant - Urysohn cellularity $Uc(X)$ instead of cellularity $c(X)$. We introduce the $n$-Urysohn cellularity $n$-$Uc(X)$ (where $ngeq 2$) and prove that the previous inequality is true in the class of $n$-Urysohn spaces replacing $Uc(X)$ by the weaker $n$-$Uc(X)$. We also show that $|X|leq 2^{Uc(X)pi chi(X)}$ if $X$ is a power homogeneous Urysohn space.