First countable, countably compact spaces and the continuum hypothesis

Abstract

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 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 .

CH and the Moore–Mrowka problem

“…We do not need the details of these definitions here because sections 4, 5, and 6 of [5] are devoted to showing that forcings of the form P[X, F ] satisfy these three conditions, and countably closed forcings satisfy them in a trivial way. The conclusion of Theorem 4 of [6] is that P ω2 is totally proper, and the result regarding quotient forcing follows from Proposition 6.13 of the same paper.…”

“…Proposition 5.2. Admissible iterations satisfy the assumptions of Theorem 4 in [6]. Thus, the limit P ω2 is totally proper, and for any α < ω 2 the quotient forcing P ω2 / Ġα is totally proper.…”

“…Let P be an admissible iteration. Theorem 4 of [6] requires the iterands of P to have three properties: they each must be totally proper, weakly < ω 1 -proper, and satisfy a complicated "iteration condition". We do not need the details of these definitions here because sections 4, 5, and 6 of [5] are devoted to showing that forcings of the form P[X, F ] satisfy these three conditions, and countably closed forcings satisfy them in a trivial way.…”

Pseudoradial spaces and copies of ω1 + 1

“…This question was also considered under PFA in [1], where other conditions for the existence of copies of ω 1 are given. Later, Eisworth and Nyikos in [7] give a model of CH in which every first countable, countably compact, noncompact space contains a copy of ω 1 . Also, in [6] Eisworth shows that any perfect pre-image of ω 1 with countable tightness contains a closed copy of ω 1 .…”

Forcing axioms and the continuum hypothesis. Part II: transcending ω1-sequences of real numbers

“…While the construction of the sequence of trees is elementary, the reader is assumed to have a solid background in set theory at the level of [5]. Knowledge of proper forcing will be required at some points, although the necessary background will be reviewed for the readers convenience; further reading can be found in [3] [4] [8]. Notation is standard and will generally follow that of [5].…”