A note on the tightness of $G_{\delta }$-modifications

Abstract

We construct a normal countably tight $T_1$ space $X$ with $t(X_{\delta })>2^{\omega }$. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss. We also show that if the continuum is not so large, then the tightness of $G_{\delta }$-modifications of countably tight spaces can be arbitrary large up to the least ${\omega }_1$-strongly compact cardinal.