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.
Citation
Ivan Gotchev. Vladimir Tkachuk. "Cardinal inequalities with Shanin number and $\pi$-character." Bull. Belg. Math. Soc. Simon Stevin 29 (3) 389 - 403, december 2022. https://doi.org/10.36045/j.bbms.220321a
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