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 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 βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) 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*.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-1,
author = {Alan Dow and Saharon Shelah},
title = {More on tie-points and homeomorphism in $\mathbb{N}$*},
journal = {Fundamenta Mathematicae},
volume = {205},
year = {2009},
pages = {191-210},
zbl = {1194.03042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-1}
}
Alan Dow; Saharon Shelah. More on tie-points and homeomorphism in ℕ*. Fundamenta Mathematicae, Tome 205 (2009) pp. 191-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm203-3-1/