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

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 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 .

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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.