  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 betaN setminus N and in the recent study of (precisely) 2-to-1 maps on betaN
setminus N . In these cases the tie-points have been the unique fixed point of
an involution on betaN setminus N. This paper is motivated by the search for
2-to-1 maps and obtaining tie-points of strikingly differing characteristics.
