First countable, countably compact spaces and the continuum hypothesis |
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| Authors: | Todd Eisworth Peter Nyikos |
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| Affiliation: | Department of Mathematics, University of Northern Iowa, Cedar Falls, Iowa 50613 ; Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 |
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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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| Keywords: | Proper forcing iterations Continuum Hypothesis pre--images of $omega_1$ |
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