Comparing Fr\'echet-Urysohn filters with two pre-orders

TL;DR

This paper investigates Fréchet-Urysohn filters on the natural numbers, distinguishing them using two pre-orders, and constructs large chains and antichains within these filters based on the Rudin-Keisler order.

Contribution

It introduces a new classification of Fréchet-Urysohn filters using two pre-orders and constructs large ordered and antichain structures among these filters.

Findings

Constructed an RK-chain of size continuum above any FU-filter.

Established the existence of an infinite RK-antichain of FU-filters.

Abstract

A filter $\F$ on $\w$ is called Fr\'echet-Urysohn if the space with only one non-isolated point $\w \cup \{\F\}$ is a Fr\'echet-Urysohn space, where the neighborhoods of the non-isolated point are determined by the elements of $\F$. In this paper, we distinguish some Fr\'echet-Urysohn filters by using two pre-orderings of filters: One is the Rudin-Keisler pre-order and the other one was introduced by Todor\v{c}evi\'c-Uzc\'ategui 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.