Tie-points and fixed-points in N^*

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

A Universal Continuum of Weight aleph

We prove that every continuum of weight aleph_1 is a continuous image of the Cech-Stone-remainder R^* of the real line. It follows that under CH the remainder of the half line [0,infty) is universal among the continua of weight c --- universal in the `mapping onto' sense. We complement this result by showing that 1) under MA every continuum of weight less than c is a continuous image of R^* 2) in the Cohen model the long segment of length omega_2+1 is not a continuous image of R^*, and 3) PFA implies that I_u is not a continuous image of R^*, whenever u is a c-saturated ultrafilter. We also show that a universal continuum can be gotten from a c-saturated ultrafilter on omega and that it is consistent that there is no universal continuum of weight c.Comment: 15 pages; 1999-01-27: revision, following referee's report; improved presentation some additional results; 2000-01-24: final version, to appear in Trans. Amer. Math. So

Far points and discretely generated spaces

We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this, we study the cardinal number given by the smallest possible character of remote and far sets of separable metrizable spaces. Finally, we prove that in some cases a countable space has far points