Cardinal inequalities with Shanin number and $\pi$-character

Abstract

It is proved that, under GCH, for any Hausdorff space $X$, we have the inequality $|X| \leq sh(X)^{\pi\chi(X) \cdot \psi_c(X)}$ and therefore $|X\leq sh(X)^{\chi(X)}$; here $sh(X)=\min\{\kappa\geq \omega: \kappa^+$ is a caliber of $X\}$ is the Shanin number of $X$. If $X$ is regular, and GCH holds, then $d(X) \leq sh(X)^{t(X)\cdot \pi\chi(X)}$. We also establish, in ZFC, that $|X| \leq wL(X)^{\CL\Delta(X)\cdot 2^{\pi\chi(X)}}$ whenever $X$ is a Urysohn space.