Asymmetric tie-points and almost clopen subsets of N*

TL;DR

This paper explores the properties of tie-points and almost clopen subsets in N*, examining their asymmetries and implications for involutions and set-theoretic constructions.

Contribution

It introduces the concept of almost clopen sets related to tie-points in N* and investigates their asymmetries and set-theoretic significance.

Findings

Analysis of asymmetries in almost clopen subsets of N*.

Connections between tie-points and involutions on N*.

Review of consistency results involving tie-points.

Abstract

A tie-point of a compact space is analogous to a cut-point: the complement of the point falls apart into two relatively clopen non-compact subsets. Set-theoretically a tie-point of N* is an ultrafilter whose dual maximal ideal can be generated by the union of two non-principal mod finite orthogonal ideals. We review some of the many consistency results that have depended on the construction of tie-points of N*. One especially important application, due to Velickovic, was to the existence of non-trivial involutions on N*. A tie-point of N* has been called symmetric if it is the unique fixed point of an involution. We define the notion of an almost clopen set to be the closure of one of the proper relatively clopen subsets of the complement of a tie-point. We explore asymmetries of almost clopen subsets of N* in the sense of how may an almost clopen set differ from its natural complementary almost clopen set.