Maximal σ-independent families
Rigid P-spaces
The Complex Stone-Weierstrass Property
The compact Hausdorff space X has the CSWP iff every subalgebra of C(X,ℂ) which separates points and contains the constant functions is dense in C(X,ℂ). Results of W. Rudin (1956) and Hoffman and Singer (1960) show that all scattered X have the CSWP and many non-scattered X fail the CSWP, but it was left open whether having the CSWP is just equivalent to being scattered. Here, we prove some general facts about the CSWP; in particular we show that if X is a compact ordered space,...
Compact scattered spaces in forcing extensions
We consider the cardinal sequences of compact scattered spaces in models where CH is false. We describe a number of models of in which no such space can have ℵ₂ countable levels.
Locally compact linearly Lindelöf spaces
There is a locally compact Hausdorff space which is linearly Lindelöf and not Lindelöf. This answers a question of Arhangel'skii and Buzyakova.
Between Martin's Axiom and Souslin's Hypothesis
Solution of Kuratowski's problem on function having the Baire property, I
Properties of the class of measure separable compact spaces
We investigate properties of the class of compact spaces on which every regular Borel measure is separable. This class will be referred to as MS. We discuss some closure properties of MS, and show that some simply defined compact spaces, such as compact ordered spaces or compact scattered spaces, are in MS. Most of the basic theory for regular measures is true just in ZFC. On the other hand, the existence of a compact ordered scattered space which carries a non-separable (non-regular) Borel measure...
Locally constant functions
Let X be a compact Hausdorff space and M a metric space. is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of as a normed linear space. We also build three first countable Eberlein...
Measures on compact HS spaces
We construct two examples of a compact, 0-dimensional space which supports a Radon probability measure whose measure algebra is isomorphic to the measure algebra of . The first construction uses ♢ to produce an S-space with no convergent sequences in which every perfect set is a . A space with these properties must be both hereditarily normal and hereditarily countably paracompact. The second space is constructed under CH and is both HS and HL.
Bohr compactifications of discrete structures
We prove the following theorem: Given a⊆ω and , if for some and all u ∈ WO of length η, a is , then a is .We use this result to give a new, forcing-free, proof of Leo Harrington’s theorem: -Turing-determinacy implies the existence of .
The power set of ω Elementary submodels and weakenings of CH
We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles. SEP is implied by each of CH*, the weak Freeze-Nation property of (ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle , but does not follow from , or even .
Some additive properties of sets of real numbers
On inhomogeneity of products of compact F-spaces
Measures on Corson compact spaces
We prove that the statement: "there is a Corson compact space with a non-separable Radon measure" is equivalent to a number of natural statements in set theory.
On the open-open game
We modify a game due to Berner and Juhász to get what we call “the open-open game (of length ω)”: a round consists of player I choosing a nonempty open subset of a space X and II choosing a nonempty open subset of I’s choice; I wins if the union of II’s open sets is dense in X, otherwise II wins. This game is of interest for ccc spaces. It can be translated into a game on partial orders (trees and Boolean algebras, for example). We present basic results and various conditions under which I or II...
Seven characterizations of non-meager 𝖯-filters
We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager -filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager -filter. Here, we identify a filter with a subspace of through characteristic functions. Along the way, we generalize to non-meager -filters a result of Miller (1984) about -points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem...
Hausdorff measures and two point set extensions
We investigate the following question: under which conditions is a σ-compact partial two point set contained in a two point set? We show that no reasonable measure or capacity (when applied to the set itself) can provide a sufficient condition for a compact partial two point set to be extendable to a two point set. On the other hand, we prove that under Martin's Axiom any σ-compact partial two point set such that its square has Hausdorff 1-measure zero is extendable.
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