Productively Lindel\"of spaces of countable tightness

Michael asked whether every productively Lindel\"of space is powerfully Lindel\"of. Building of work of Alster and De la Vega, assuming the Continuum Hypothesis, we show that every productively Lindel\"of space of countable tightness is powerfully Lindel\"of. This strengthens a result of Tall and Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a productively Lindel\"of space $X$ is powerfully Lindel\"of if every open cover of $X^\omega$ admits a point-continuum refinement consisting of basic open sets. This strengthens a result of Burton and Tall. Finally, we show that separation axioms are not relevant to Michael's question: if there exists a counterexample (possibly not even $\mathsf{T}_0$), then there exists a regular (actually, zero-dimensional) counterexample.Comment: 7 page

GδG_\delta-topology and compact cardinals

For a topological space $X$, let $X_\delta$ be the space $X$ with $G_\delta$-topology of $X$. For an uncountable cardinal $\kappa$, we prove that the following are equivalent: (1) $\kappa$ is $\omega_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindel\"of degree of $X_\delta$ is $\le \kappa$. (3) For every compact Hausdorff space $X$, the weak Lindel\"of degree of $X_\delta$ is $\le \kappa$. This shows that the least $\omega_1$-strongly compact cardinal is the supremum of the Lindel\"of and the weak Lindel\"of degrees of compact Hausdorff spaces with $G_\delta$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_\delta$-topology. For the square of a Lindel\"of space, using weak $G_\delta$-topology, we prove that the following are consistent: (1) the least $\omega_1$-strongly compact cardinal is the supremum of the (weak) Lindel\"of degrees of the squares of regular $T_1$ Lindel\"of spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindel\"of spaces