"Cardinal inequalities with Shanin number and $\pi$-character"Gotchev, IvanWe recall that a regular cardinal number $\kappa$ is a caliber of a topological space $X$ if for any family $\mathcal{U}$ of non-empty open subsets of $X$ such that $|\mathcal{U}|=\kappa$, there exists a family $\mathcal{V}\in [\mathcal{U}]^\kappa$ such that $\bigcap\mathcal{V}\ne \emptyset$. The cardinal number $sh(X) = \min\{\kappa \ge \omega : \kappa^+$ is a caliber of $X\}$ is called the Shanin number of the space $X$. It is easy to see that $c(X)\le sh(X)\le d(X)$ for any space $X$. In this talk we will present some cardinal inequalities involving the Shanin number and the $\pi$-character $\pi\chi(X)$ of a space $X$. Among other results we will show that, under GCH, for every Hausdorff space $X$ we have $|X|\le sh(X)^{\pi\chi(X)\psi_c(X)}$ and therefore $|X|\le sh(X)^{\chi(X)}$. These give, under GCH, formal generalizations of the Willard--Dissananyake's inequality $|X|\le d(X)^{\pi\chi(X)\psi_c(X)}$ and of Pospi\v{s}il's inequality $|X|\le d(X)^{\chi(X)}$, which are true for every Hausdorff space $X$. In addition, if $X$ is a regular $T_1$-space, and GCH holds, then $d(X)\le sh(X)^{\pi\chi(X)t(X)}$. This is a joint work with Vladimir Tkachuk. |
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