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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A universal continuum of weight $\aleph$


Authors: Alan Dow and Klaas Pieter Hart
Journal: Trans. Amer. Math. Soc. 353 (2001), 1819-1838
MSC (1991): Primary 54F15; Secondary 03E35, 04A30, 54G05
Published electronically: June 20, 2000
MathSciNet review: 1707489
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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  $\mathsf{CH}$ 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 $R^*$, 2) in the Cohen model the long segment of length  $\omega_2+1$ is not a continuous image of $R^*$, and 3)  $\mathsf{PFA}$ implies that $I_u$ is not a continuous image of $R^*$, whenever $u$ 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  $\mathfrak{c}$.


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Additional Information

Alan Dow
Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
Email: dowa@yorku.ca

Klaas Pieter Hart
Affiliation: Faculty of Technical Mathematics and Informatics, TU Delft, Postbus 5031, 2600 GA Delft, The Netherlands
Email: k.p.hart@twi.tudelft.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02601-5
PII: S 0002-9947(00)02601-5
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
Received by editor(s): October 10, 1996
Received by editor(s) in revised form: January 14, 1999
Published electronically: June 20, 2000
Additional Notes: The research of the second author was supported by The Netherlands Organization for Scientific Research (NWO) — Grant R61-322
Article copyright: © Copyright 2000 American Mathematical Society