On a problem of Angelo Bella

Abstract

The main result of this note is the following theorem. "If $X$ is any Hausdorff space with $\kappa =\hat{F}(X)\cdot \hat{\mu }(X)$ then $L(X_{<\kappa })\le \varrho (\kappa )$". Here $\hat{F}(X)$ is the smallest cardinal $\phi$ so that $|S|<\phi$ for any set $S$ that is free in $X$ and $\hat{\mu }(X)$ 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 Lindelöf 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.

4 pages