A universal continuum of weight $\aleph$
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- by Alan Dow and Klaas Pieter Hart PDF
- Trans. Amer. Math. Soc. 353 (2001), 1819-1838 Request permission
Abstract:
We prove that every continuum of weight $\aleph _1$ is a continuous image of the Čech-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 $\mathfrak {c}$.References
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Additional Information
- Alan Dow
- Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3
- MR Author ID: 59480
- 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
- 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 R 61-322
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 1819-1838
- MSC (1991): Primary 54F15; Secondary 03E35, 04A30, 54G05
- DOI: https://doi.org/10.1090/S0002-9947-00-02601-5
- MathSciNet review: 1707489