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. Tie-points have appeared in the construction of
non-trivial autohomeomorphisms of beta N setminus
N= N^* and in the recent study 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^* .
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