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

Volume 203 / 2009

Alan Dow, Saharon Shelah Fundamenta Mathematicae 203 (2009), 191-210 MSC: 03A50, 54A25, 54D35. DOI: 10.4064/fm203-3-1


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}^*$.


  • Alan DowUniversity of North Carolina at Charlotte
    Charlotte, NC 28223, U.S.A.
  • Saharon ShelahDepartment of Mathematics
    Rutgers University
    Hill Center
    Piscataway, NJ 08854-8019, U.S.A.
    Institute of Mathematics
    Hebrew University
    Givat Ram, Jerusalem 91904, Israel

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