First Countable Continua and Proper Forcing
نویسندگان
چکیده
Assuming the Continuum Hypothesis, there is a compact first countable connected space of weight א1 with no totally disconnected perfect subsets. Each such space, however, may be destroyed by some proper forcing order which does not add reals.
منابع مشابه
Subcomplete Forcing and L-Forcing
ABSRACT In his book Proper Forcing (1982) Shelah introduced three classes of forcings (complete, proper, and semi-proper) and proved a strong iteration theorem for each of them: The first two are closed under countable support iterations. The latter is closed under revised countable support iterations subject to certain standard restraints. These theorems have been heavily used in modern set th...
متن کاملNew reals: Can live with them, can live without them
Judah and Shelah [3] proved that countable support iterations of proper forcings preserve the ω-bounding property (see 2.2 here). In his book Proper and Improper Forcing [8, XVIII §3] Shelah gave several cases of general preservation theorems for proper countable support iterations (the proofs tend to be hard to digest, though). In this paper we deal with “Case A”. A simplified version of this ...
متن کاملPreservation theorems for countable support forcing iterations
A major theme of [12] is preservation theorems for iterated forcing. These are theorems of the form “if 〈Pξ : ξ ≤ κ〉 is a countable support forcing iteration based on 〈Q̇ξ : ξ < κ〉 and each Q̇ξ has property such-and-such then Pκ has property thus-and-so.” The archetypal preservation theorem is the Fundamental Theorem of Proper Forcing [12, chapter III], which states that if each Q̇ξ is proper in V...
متن کاملBounded Forcing Axioms and Baumgartner’s Conjecture
We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of αproper forcings. Following ideas of Shelah we separate them for distinct countable indec...
متن کاملBaumgartner's conjecture and bounded forcing axioms
We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in...
متن کامل