Regular $G_δ$-diagonals and some upper bounds for cardinality of topological spaces

Ivan S. Gotchev, Mikhail G. Tkachenko, Vladimir V. Tkachuk

Published 2015-06-15Version 1

We recall that for a Urysohn space $X$ the \emph{regular diagonal degree $\overline{\Delta}(X)$ of $X$} is defined as the minimal infinite cardinal $\kappa$ such that $X$ has a regular $G_\kappa$-diagonal. The \emph{o-tightness} of $X$ does not exceed $\kappa$, or $\mathrm{ot}(X)\le\kappa$, if for every family $\mathcal{V}$ of open subsets of $X$ and for every point $x\in X$ with $x\in\overline{\bigcup \mathcal{V}}$ there exists a subfamily $\mathcal{V}\subset \mathcal{U}$ such that $|\mathcal{V}|\le\kappa$ and $x\in\overline{\bigcup\mathcal{V}}$. We will say that the \emph{dense o-tightness} of $X$ does not exceed $\kappa$, or $\mathrm{dot}(X)\le\kappa$, if for every family $\mathcal{U}$ of open subsets of $X$ whose union is dense in $X$ and for every point $x\in X$ there exists a subfamily $\mathcal{V}\subset\mathcal{U}$ such that $|\mathcal{V}|\le\kappa$ and $x\in\overline{\bigcup\mathcal{V}}$. The main results of this paper are as follows: If $X$ is a Urysohn space then (1) $|X|\le wL(X)^{\pi\chi(X)\cdot\overline{\Delta}(X)}$ and (2) $|X|\le wL(X)^{s\Delta_2(X)\cdot{\mathrm{dot}(X)}}$; (3) if $2^\omega=\omega_1$ and $X$ is a space with a regular $G_\delta$-diagonal and caliber $\omega_1$ then $X$ is separable; if $X$ is a Hausdorff space then (4) $|X|\le\pi w(X)^{\mathrm{ot}(X)\cdot\psi_c(X)}$ and (5) $|X|\le \pi\chi(X)^{\mathrm{ot}(X)\cdot\psi_c(X)\cdot aL_c(X)}$. Inequalities (1) and (2) generalize some resent results obtained by I. S. Gotchev and D. Basile, A. Bella, and G. J. Ridderbos. As a corollary of (3) we obtain (under CH) a generalization and a positive answer respectively of a result and a question of R. Buzyakova. We use (4) to prove (5) which gives another generalization of the famous Arhangel'skii's inequality that the cardinality of any Hausdorff space $X$ does not exceed $2^{\chi(X)\cdot L(X)}$.