Generalized Mrówka spaces and diagonal properties

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].