A Bound for the Density of Any Hausdorff Space

Cardinal functions and cardinal inequalities on topological spaces continue to be an active area of research in set-theoretic topology. Arhangel’skii’s 1969 result that a compact, first countable Hausdorff space has cardinality at most $\mathfrak{c}$, the cardinality of the continuum, continues to inspire research in recent decades, as well as the Hajnal-Juhász theorem that a first countable Hausdorff space with the countable chain condition has the same upper bound.
In this talk we will give a brief introduction to the topic and launch into some recent results of the speaker. We show, in a certain specific sense, that both the density and the cardinality of a Hausdorff space are related to the degree to which the space is nonregular. It was shown by Sapirovskii that $d(X)\leq\pi\chi(X)^{c(X)}$ for a regular space $X$ and the speaker observed this holds if the space is only quasiregular. We generalize this result to the class of all Hausdorff spaces by introducing the nonquasiregularity degree $nq(X)$, which is countable when $X$ is quasiregular, and showing $d(X)\leq\pi\chi(X)^{c(X)nq(X)}$ for any Hausdorff space $X$. This demonstrates that the degree to which a space is nonquasiregular has a fundamental and direct connection to its density and, ultimately, its cardinality. Importantly, if $X$ is Hausdorff then $nq(X)$ is "small" in the sense that $nq(X)\leq\min\{\psi_c(X),L(X),pct(X)\}$. This results in a unified proof of both Sapirovskii's density bound for regular spaces and Sun’s bound $\pi\chi(X)^{c(X)\psi_c(X)}$ for the cardinality of a Hausdorff space $X$. A consequence is an improved bound for the cardinality of a Hausdorff space. We give an example of a compact, Hausdorff space $X$ for which our new bound is a strict improvement over Sun's bound, which in turn is an improvement over the Hajnal-Juhász theorem mentioned above.