More on Tie-points and homeomorphism in N^*

TL;DR

This paper explores the concept of tie-points in N^*, their role in constructing autohomeomorphisms and 2-to-1 maps, and demonstrates the consistency of certain non-homeomorphic 2-to-1 images of N^* in set-theoretic topology.

Contribution

It introduces new properties of tie-points in N^* and shows the consistency of non-homeomorphic 2-to-1 continuous images of N^*.

Findings

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

Existence of 2-to-1 continuous images of N^* that are not homeomorphic to N^*.

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= N^* and in the recent study of (precisely) 2-to-1 maps on N^*. In these cases the tie-points have been the unique fixed point of an involution on N^*. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of N^* which is not a homeomorph of N^* .