Some remarks on generalized Martin's axiom

Z. Spasojevi\'c

Abstract: Let $GMA$ denote that if ${\Bbb P}$ is well-met, strongly $\omega_1$-closed and $\omega_1$-centered partial order and ${\Cal D}$ a family of $<2^{\omega_1}$ dense subsets of ${\Bbb P}$: then there is a filter $G\subseteq {\Bbb P}$ which meets every member of ${\Cal D}$. The consistency of $2^\omega = \omega_1 + 2^{\omega_1}>\omega_2 + GMA$ was proved by Baumgartner [1] and in [13] many of its consequences were considered. In this paper we give a consequence and present an independence result. Namely, we prove that, as a consequence of $2^\omega = \omega_1 + 2^{\omega_1}>\omega_2 + GMA$, every $\leq^*$-increasing $\omega_2$-sequence in $(\omega_1^{\omega_1},\leq^*)$ is a lower half of some $(\omega_2,\omega_2)$-gap and show that the existence of an $\omega_2$-Kurepa tree is consistent with and independent of $2^\omega = \omega_1 + 2^{\omega_1}>\omega_2 + GMA$.

Classification (MSC2000): 03E35

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