Normal filters generated by a family of sets

Di Prisco, Carlos Augusto; Uzcategui, Carlos Enrique

doi:10.1090/S0002-9939-1987-0908659-0

Normal filters generated by a family of sets
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by Carlos Augusto Di Prisco and Carlos Enrique Uzcategui PDF

Proc. Amer. Math. Soc. 101 (1987), 513-518 Request permission

Abstract:

We study normal filters on the set spaces $\lambda ,{\mathcal {P}_\kappa }\left ( \lambda \right ),{\left [ \lambda \right ]^\kappa }$, and ${\left ( \lambda \right )^\kappa }$. We characterize the least normal $\gamma$-complete filter containing a given $\gamma$-complete filter for $\gamma \geq {\omega _1}$. If $\mathcal {F}$ is a ${\omega _1}$-complete filter on any of the set spaces mentioned, the least ${\omega _1}$-complete normal filter containing it is the filter generated by the sets $\left \{ {x \in E\left | {{\alpha _1}, \ldots ,{\alpha _n} \in x \to x \in f\left ( {{\alpha _1}, \ldots ,{\alpha _n}} \right )} \right .} \right \}$ where $f:{\lambda ^{ < \omega }} \to \mathcal {F}$.

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