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arXiv · 1506.04665

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

Ivan S. Gotchev·Mikhail G. Tkachenko·Vladimir V. Tkachuk
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Abstract

We prove that, under CH, any space with a regular $G_δ$-diagonal and caliber $ω_1$ is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space $X$, we establish the inequality $|X|\le wL(X)^{sΔ_2(X)\cdot{dot(X)}}$ which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if $X$ is a Hausdorff space, then $|X|\le(πχ(X)\cdot d(X))^{ot(X)\cdotψ_c(X)}$; this result implies Šapirovski{\uı}'s inequality $|X|\leπχ(X)^{c(X)\cdotψ(X)}$ which only holds for regular spaces. It is also proved that $|X|\le πχ(X)^{ot(X)\cdotψ_c(X)\cdot aL_c(X)}$ for any Hausdorff space $X$; this gives one more generalization of the famous Arhangel$^\prime$skii's inequality $|X|\le 2^{χ(X)\cdot L(X)}$.

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Ivan S. Gotchev, Mikhail G. Tkachenko, Vladimir V. Tkachuk. 2016-02-26. Regular $G_δ$-diagonals and some upper bounds for cardinality of topological spaces. https://arxiv.org/abs/1506.04665

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