Tie-points and fixed-points in N^*

TL;DR

This paper explores the concept of tie-points in the space N* and their role in constructing non-trivial automorphisms and 2-to-1 maps on betaN setminus N, aiming to find diverse tie-point characteristics.

Contribution

It introduces the study of tie-points in N*, focusing on their properties and potential for creating novel 2-to-1 maps and automorphisms of betaN setminus N.

Findings

Tie-points can be used to construct non-trivial autohomeomorphisms.

Tie-points serve as fixed points for involutions on betaN setminus N.

The paper identifies conditions for the existence of diverse tie-points.

Abstract

A point x is a (bow) tie-point of a space X if X setminus {x} can be partitioned into (relatively) clopen sets each with x in its closure. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of betaN setminus N and in the recent study of (precisely) 2-to-1 maps on betaN setminus N . In these cases the tie-points have been the unique fixed point of an involution on betaN setminus N. This paper is motivated by the search for 2-to-1 maps and obtaining tie-points of strikingly differing characteristics.