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First countable, countably compact spaces and the continuum hypothesis


Authors: Todd Eisworth and Peter Nyikos
Journal: Trans. Amer. Math. Soc. 357 (2005), 4269-4299
MSC (2000): Primary 03E75
DOI: https://doi.org/10.1090/S0002-9947-05-04034-1
Published electronically: June 21, 2005
MathSciNet review: 2156711
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Abstract: We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy of $\omega_1$ with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah's iteration theorems appearing in Chapters V and VIII of Proper and improper forcing (1998), as well as Eisworth and Roitman's (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah's club-guessing sequences) that shows similar results do not hold for closed pre-images of $\omega_2$.


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Additional Information

Todd Eisworth
Affiliation: Department of Mathematics, University of Northern Iowa, Cedar Falls, Iowa 50613
Address at time of publication: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: eisworth@uni.edu, eisworth@math.ohiou.edu

Peter Nyikos
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: nyikos@math.sc.edu

DOI: https://doi.org/10.1090/S0002-9947-05-04034-1
Keywords: Proper forcing, iterations, Continuum Hypothesis, pre--images of $\omega_1$
Received by editor(s): May 23, 2002
Published electronically: June 21, 2005
Additional Notes: The first author was partially supported by a Summer Fellowship granted by the Graduate College of the University of Northern Iowa
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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