$G_\delta$-modification of compacta and cardinal invariants

summary:Given a space $X$, its $G_\delta $-subsets form a basis of a new space $X_\omega $, called the $G_\delta $-modification of $X$. We study how the assumption that the $G_\delta $-modification $X_\omega $ is homogeneous influences properties of $X$. If $X$ is first countable, then $X_\omega $ is discrete and, hence, homogeneous. Thus, $X_\omega $ is much more often homogeneous than $X$ itself. We prove that if $X$ is a compact Hausdorff space of countable tightness such that the $G_\delta $-modification of $X$ is homogeneous, then the weight $w(X)$ of $X$ does not exceed $2^\omega $ (Theorem 1). We also establish that if a compact Hausdorff space of countable tightness is covered by a family of $G_\delta $-subspaces of the weight $\leq c=2^\omega $, then the weight of $X$ is not greater than $2^\omega $ (Theorem 4). Several other related results are obtained, a few new open questions are formulated. Fedorchuk's hereditarily separable compactum of the cardinality greater than $c=2^\omega $ is shown to be $G_\delta $-homogeneous under CH. Of course, it is not homogeneous when given its own topology