Perfectly normal nonrealcompact spaces under Martin's Maximum

Abstract

We analyze the behavior of a perfectly normal nonrealcompact space $({\omega }_1,\tau )$ on ${\omega }_1$ such that for every , $\gamma$ is $\tau$-open and $\gamma +\omega$ is $\tau$-closed under Martin's Maximum. We show that there exists a club subset $D$ of ${\omega }_1$ such that for a stationary subset of $\delta \in acc\left(D\right)$, for all $\tau$-open neighborhood $N$ of $\delta +n$, there exists such that for all $\xi \in D\cap [\eta ,\delta )$, $N\cap \xi$ is unbounded in $\xi$.