Normal spaces whose Stone-Čech remainders have countable tightness

Abstract:

We prove, assuming PFA, that each normal space whose Stone-Čch remainder has countable tightness is ACRIN. A normal space $X$ is called ACRIN if each of its regular images is normal. Fleissner and Levy proved that if $X$ is normal and every countably compact subset of the Stone-Čech remainder $\beta X\backslash X$ is closed in $\beta X\backslash X$, then $X$ is ACRIN. They asked if each normal space whose Stone-Čech remainder has countable tightness is ACRIN. Theorem $2$ gives the positive answer assuming the Proper Forcing Axiom.