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

Alan Dow ¹   ; Saharon Shelah ²

¹ University of North Carolina at Charlotte Charlotte, NC 28223, U.S.A.
² Department of Mathematics Rutgers University Hill Center Piscataway, NJ 08854-8019, U.S.A. and Institute of Mathematics Hebrew University Givat Ram, Jerusalem 91904, Israel

Fundamenta Mathematicae, Tome 203 (2009) no. 3, pp. 191-210

Résumé

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}^*$.

Détail

DOI : 10.4064/fm203-3-1

Keywords: point bow tie point space setminus partitioned relatively clopen sets each its closure denote mathbin mathop bowtie limits where closed sets which have unique common accumulation point tie points have appeared construction non trivial autohomeomorphisms beta mathbb mathbb * veli kovi shelah amp stepr recent study levy dow amp techanie precisely to maps mathbb * these cases tie points have unique fixed point involution mathbb * application results paper consistency there being to continuous image mathbb * which homeomorph nbsp mathbb *