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A universal continuum of weight

Authors:	Alan Dow Klaas Pieter Hart

Institution:	Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 ; Faculty of Technical Mathematics and Informatics, TU Delft, Postbus 5031, 2600 GA Delft, The Netherlands

Abstract:	We prove that every continuum of weight $\aleph_1$ is a continuous image of the Cech-Stone-remainder of the real line. It follows that under $\mathsf{CH}$ the remainder of the half line $0,\infty)$ is universal among the continua of weight $\mathfrak{c}$ -- universal in the `mapping onto' sense. We complement this result by showing that 1) under $\mathsf{MA}$ every continuum of weight less than $\mathfrak{c}$ is a continuous image of , 2) in the Cohen model the long segment of length $\omega_2+1$ is not a continuous image of , and 3) $\mathsf{PFA}$ implies that is not a continuous image of , whenever is a $\mathfrak{c}$ -saturated ultrafilter. We also show that a universal continuum can be gotten from a $\mathfrak{c}$ -saturated ultrafilter on $\omega$ , and that it is consistent that there is no universal continuum of weight $\mathfrak{c}$ .

Keywords:	Parovi\v{c}enko's theorem universal continuum remainder of $[0 \infty)$ $\aleph_1$-saturated model elementary equivalence Continuum Hypothesis Cohen reals long segment Martin's Axiom Proper Forcing Axiom saturated ultrafilter

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