### Asymmetric tie-points on almost clopen subsets of $\mathbb{N}^*$

by Dow and Shelah. [DwSh:1057]

Commentationes Math Universitatis Carolinae, 2018

A tie-point of compact space is analogous to a cut-point: the
complement of the point falls apart into two relatively clopen
non-compact subsets. We review some of the many consistency results
that have depended on the construction of tie-points of {N}^* .
One especially important application, due to Velickovic, was to the
existencce of non-trivial involutions on {N}^* . A tie-point
{N}^* has been called symmetric if it is the unique fixed
point of an involution. We define the notion of almost clopen set to
be the closure of one of the proper relatively clopen subsets of the
complement of a tie-point. We explore asymmetries of almost clopen
subsets of {N}^* in the sense of how may an almost clopen
set differ from its natural complementary almost clopen set.

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