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may 2020 Generalized Mrówka spaces and diagonal properties
Wei-Feng Xuan, Yan-Kui Song, Wei-Xue Shi
Bull. Belg. Math. Soc. Simon Stevin 27(1): 61-69 (may 2020). DOI: 10.36045/bbms/1590199304

Abstract

In this paper, we prove that if $2^\lambda < \kappa$ and $\mathcal A \subset [\kappa]^{\lambda}$ is a MADF then the generalized Mrówka space $\Psi (\mathcal A\setminus \mathcal B)$ has no $G_\lambda$-diagonal, where $\mathcal B \subset \mathcal A$ and $|\mathcal B|\le\lambda$. If a topological space $X$ has a regular $G_\lambda$-diagonal and a local decreasing base of cardinality $\chi(X)$ for each $x \in X$, then $|X| \le 2^{dc(X) \cdot \chi(X) \cdot \lambda}$, where $dc(X)$ is the discrete cellularity of $X$ and $\chi(X)$ is the character of $X$. As a corollary, we prove that if $X$ is a DCCC (and hence, $DC(\omega_1)$, weakly Lindelöf or star Lindelöf) first countable space with a regular $G_\delta$-diagonal then $|X|\le 2^\omega$, which gives a positive answer to a question in [17]. Finally, we give another counterexample to a question of Ginsburg and Woods [6], which has nicer properties than the counterexamples constructed in [15] and [16].

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Wei-Feng Xuan. Yan-Kui Song. Wei-Xue Shi. "Generalized Mrówka spaces and diagonal properties." Bull. Belg. Math. Soc. Simon Stevin 27 (1) 61 - 69, may 2020. https://doi.org/10.36045/bbms/1590199304

Information

Published: may 2020
First available in Project Euclid: 23 May 2020

zbMATH: 07213658
MathSciNet: MR4102701
Digital Object Identifier: 10.36045/bbms/1590199304

Subjects:
Primary: 54D20
Secondary: 54E35

Rights: Copyright © 2020 The Belgian Mathematical Society

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Vol.27 • No. 1 • may 2020
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