## More on tie-points and homeomorphism in $\mathbb N^*$

### Volume 203 / 2009

#### 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. We denote this as $X = A \mathbin{\mathop{\bowtie}\limits_{x}} B$ where $A, B$ are the closed sets which have a unique common accumulation point $x$. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of $\beta{\mathbb N}={\mathbb N}^*$ (by Veličković and Shelah & Stepr{# ma}ns) and in the recent study (by Levy and Dow & Techanie) of precisely 2-to-1 maps on ${\mathbb N}^*$. In these cases the tie-points have been the unique fixed point of an involution on ${\mathbb N}^* $. One application of the results in this paper is the consistency of there being a 2-to-1 continuous image of ${\mathbb N}^*$ which is not a homeomorph of $ {\mathbb N}^*$.