On the cardinality of Hausdorff spaces and H-closed spaces

Abstract

We introduce the cardinal invariant $a{L}^{\prime }(X)$ and show that $|X|\le 2^{a{L}^{\prime }(X)\chi (X)}$ for any Hausdorff space $X$ (a corollary of Theorem 4.4. This invariant has the properties a) $a{L}^{\prime }(X)={\aleph }_0$ if $X$ is H-closed, and b) $aL(X)\le a{L}^{\prime }(X)\le a{L}_c(X)$. Theorem 4.4 then gives a new improvement of the well-known Hausdorff bound $2^{L(X)\chi (X)}$ from which it follows that $|X|\le 2^{{\psi }_c(X)}$ if $X$ is H-closed (Dow/Porter [5]). The invariant $a{L}^{\prime }(X)$ is constructed using convergent open ultrafilters and an operator $c:\scr{P}(X)\to\scr{P}(X)$ with the property $clA\subseteq c(A)\subseteq c{l}_{\theta }(A)$ for all $A\subseteq X$. As a comparison with this open ultrafilter approach, in $\S 3$ we additionally give a $\kappa$-filter variation of Hodel's proof [10] of the Dow-Porter result. Finally, for an infinite cardinal $\kappa$, in $\S 5$ we introduce $\kappa$wH-closed spaces, $\kappa H^{\prime }$-closed spaces, and $\kappa H^{\prime \prime }$-closed spaces. The first two notions generalize the H-closed property. Key results in this connection are that a) if $\kappa$ is an infinite cardinal and $X$ a $\kappa$wH-closed space with a dense set of isolated points such that $\chi (X)\le \kappa$, then $|X|\le 2^{\kappa }$, and b) if $X$ is $\kappa H^{\prime }$-closed or $\kappa H^{\prime \prime }$-closed then $a{L}^{\prime }(X)\le \kappa$. This latter result relates these notions to the invariant $a{L}^{\prime }(X)$ and the operator $c$.

arXiv comment22 pages