More on tie-points and homeomorphism in N*

Abstract

A point x is a (bow) tie-point of a space X if X \ {x} can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as X = A?B where A, B are the closed sets which have a unique common accumulation point x. T?ie-points have appeared in the construction of non-trivial autohomeomorphisms of βN\N = N* (by Veličković and Shelah & Steprāns) and in the recent study (by Levy and Dow & Techanie) of precisely 2-to-l 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-l continuous image of N* which is not a homeomorph of N*. © Instytut Matematyczny PAN, 2009.