Coinitial families of perfect sets

M. Balcerzak and A. Roslanowski

Abstract: Let $\hbox{Perf}(X)$ denote the family of all perfect subsets of a perfect Polish space $X$. We show several properties and characterizations of coinitial subfamilies of $\hbox{Perf}(X)$. Connections with the Vietoris topology and Morgan's category bases are observed. We characterize the least cardinality of a family of coinitial ideals with noncoinitial intersection. Some examples of special noncoinitial families in $\hbox{Perf}(X)$ are presented. Much place is devoted to perfect isomorphisms between ideals of subsets of $X$.

Keywords: Polish space, perfect set, $\so$-set, coinitial set, ideal of sets,isomorphism of ideals, category base, Vietoris topology, Sacks Amoeba forcing

Classification (MSC2000): 28A05, 06A07, 04A15, 54E50, 54H05, 54B20

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