On the Origin of New Compact Spaces in Forcing Models

Abstract

Within the past 10 -15 years, many concrete problems of set-theoretical topology coulcl by solved by forcing (see [ 6 ] , [6]). To prove the consistency of the existence of topological spaces satisfying specific properties, it is shown that for a suitable p~~rtiiilly ordered set P, there exists a space 20 Tvith the desired properties in the generic extension &! [ GI . I n forcing proofs of particular problems, diverse combinatorinl properties of P in the ground model X play a decisive role. This paper considers more general connections between new cornpact spaces in &! [ GI and certain mays in M . We show, that not an abstract construction, but a topological object in M is responsible for the existence of 20. In 111 there is an open continuous mapping f : Z-X from a compact space Z onto a compact space X, where the Booman algebra of regular open sets in X contains n dense subset isomorphic to P. f can be reconstructed in M [ G ] and G defines n point r o € X . We then have: Zo=f-l{ro}.Notation. A , B and C denote BooLmn algebras. We use 0, 1, A , V , -for the zero, the unit and the Booman operations, respectively, and s for the partial order. An embedding of A in B is a n isomorphism of A onto a subalgebra of B..A subalgebra C of B is said to be dense in B, if for every nonzero U c B there is a nonzero V c C such that V s U . An embedding such that the range is dense is called a dense embedding. A denotes the completion of Booman algebra A .All topological spaces under consideration in this paper are HAUSDORFF spaces. A11 mappings of topological spaces are continuous. By R O ( X ) we denote the BooLEan algebra of regular open subsets of the space X. Some facts about forcing. Set theory means ZERJIELO-FRAENKEL set theory with axiom of choice. All models of set theory are cmumed t o be transitive.Let &I be a model of set theory, and let B be a Booman algebra in iM. If G is B-generic over M , then N [ G ] denotes the generic extension of M (see [-I]). We

First Countable Continua and Proper Forcing

“…Note that if Y itself does not contain a Cantor subset, then (1) is trivial, and we may take all D q = ∅ to satisfy (2) and (3). Now, in our construction, we also build D q p for q ≤ p ∈ 2 <ω 1 satisfying: R10.…”

“…We get the point of uncountable character in the natural corresponding compact space X in V [G]. This compact space determined by X was described by Bandlow [1] (and later in [3,4,6]), and can be defined as follows:…”