A Universal Continuum of Weight aleph

Abstract

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.

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. Soc

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