First countable, countably compact spaces and the continuum hypothesis

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 ω 1 \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 ω 2 \omega _2 .