Perfectly normal nonrealcompact spaces under Martin's Maximum

• Published in: Mathematical logic quarterly Vol. 70; no. 4; pp. 388 - 397
• Main Author: Ishiu, Tetsuya
• Format: Journal Article
• Language: English
• Published: Berlin Wiley Subscription Services, Inc 01.11.2024
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.202400002

We analyze the behavior of a perfectly normal nonrealcompact space (ω1,τ)$(\omega _1, \tau)$ on ω1$\omega _1$ such that for every γ<ω1$\gamma <\omega _1$, γ$\gamma$ is τ$\tau$‐open and γ+ω$\gamma +\omega$ is τ$\tau$‐closed under Martin's Maximum. We show that there exists a club subset D$D$ of ω1$\omega _1$ such that for a stationary subset of δ∈acc(D)$\delta \in \operatorname{acc}(D)$, for all τ$\tau$‐open neighborhood N$N$ of δ+n$\delta +n$, there exists η<δ$\eta <\delta$ such that for all ξ∈D∩[η,δ)$\xi \in D\cap [\eta, \delta)$, N∩ξ$N\cap \xi$ is unbounded in ξ$\xi$.