Comparing Fréchet-Urysohn filters with two pre-orders

Abstract

A filter $\F$ on $\w$ is called Fréchet-Urysohn if the space with only one non-isolated point $\w \cup \{\F\}$ is a Fréchet-Urysohn space, where the neighborhoods of the non-isolated point are determined by the elements of $\F$. In this paper, we distinguish some Fréchet-Urysohn filters by using two pre-orderings of filters: One is the Rudin-Keisler pre-order and the other one was introduced by Todorčević-Uzcátegui in \cite{tu05}. In this paper, we construct an $RK$-chain of size $\c^+$ which is $RK$-above of avery $FU$-filter. Also, we show that there is an infinite $RK$-antichain of $FU$-filters.