On a problem of Angelo Bella

TL;DR

This paper proves a new upper bound on the Lindelöf number of certain topological spaces based on their free set and cover properties, providing a partial answer to a question posed by Angelo Bella.

Contribution

It introduces a novel theorem relating Lindelöf numbers to free set and cover cardinal invariants, advancing understanding of linearly Lindelöf spaces.

Findings

Established an upper bound for the Lindelöf number in terms of cardinal invariants.

Derived a condition under which the Lindelöf number of a space's G_{< κ}-modification is at most continuum.

Provided a consistent affirmative answer to a question of Angelo Bella.

Abstract

The main result of this note is the following theorem. "If $X$ is any Hausdorff space with $\kappa = \widehat{F}(X) \cdot \widehat{\mu}(X)$ then $L(X_{< \kappa}) \le \varrho(\kappa)$". Here $\widehat{F}(X)$ is the smallest cardinal $\varphi$ so that $|S| < \varphi$ for any set $S$ that is free in $X$ and $\widehat{\mu}(X)$ is the smallest cardinal $\mu$ so that, for every set $S$ that is free in $X$, any open cover of $\overline {S}$ has a subcover of size $< \mu$. Moreover, $X_{< \kappa}$ is the $G_{< \kappa}$-modification of $X$ and $\varrho(\kappa) = \min \{\varrho : \varrho ^{< \kappa} = \varrho\}$. As a corollary we obtain that if $X$ is a linearly Lindel\"of regular space of countable tightness then $L(X_\delta) \le \mathfrak{c}$, provided that $ \mathfrak{c} = 2^{< \mathfrak{c}}$. This yields a consistent affirmative answer to a question of Angelo Bella.