We consider the cardinal sequences of compact scattered spaces in models where CH is false. We describe a number of models of $2^{ℵ₀} = ℵ₂$ in which no such space can have ℵ₂ countable levels.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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, then X has the CSWP iff X does not contain a copy of the Cantor set. This provides a class of non-scattered spaces with the CSWP. Among these is the double arrow space of Aleksandrov and Urysohn. The CSWP for this space implies a Stone-Weierstrass property for the complex regulated functions on the unit interval.
3
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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 𝓟(ω), and the (ℵ₁,ℵ₀)-ideal property. SEP implies the principle $C₂^{s}(ω₂)$, but does not follow from $C₂^{s}(ω₂)$, or even $C^{s}(ω₂)$.
4
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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 is equivalent to the existence of a real-valued measurable cardinal ≤${\ninegot c}$. We show that not being in MS is preserved by all forcing extensions which do not collapse $ω_1$, while being in MS can be destroyed even by a ccc forcing.
5
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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 $2^{ω_1}$. The first construction uses ♢ to produce an S-space with no convergent sequences in which every perfect set is a $G_δ$. 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.
6
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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.
7
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
Let X be a compact Hausdorff space and M a metric space. $E_0(X,M)$ 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 $E_0(X,M)$ is all of C(X,M). These include βℕ\ℕ, 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 $E_0(X,M)$ as a normed linear space. We also build three first countable Eberlein compact spaces, F,G,H, with various $E_0$ properties. For all metric M, $E_0(F,M)$ contains only the constant functions, and $E_0(G,M) = C(G,M)$. If M is the Hilbert cube or any infinite-dimensional Banach space, then $E_0(H,M) ≠ C(H,M)$, but $E_0(H,M) = C(H,M)$ whenever $M ⊆ ℝ^n$ for some finite n.
8
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We prove the following theorem: Given a⊆ω and $1 ≤ α < ω_1^{CK}$, if for some $η < ℵ_1$ and all u ∈ WO of length η, a is $Σ _α^0(u)$, then a is $Σ_α^0$.} We use this result to give a new, forcing-free, proof of Leo Harrington's theorem: {$Σ_1^1 $-Turing-determinacy implies the existence of $0^{#}$}.
9
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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 does or does not have a winning strategy. We investigate the games on trees and Boolean algebras in detail, completely characterizing the game for $ω_1$-trees. An undetermined game is also defined. (In contrast, it is still open whether there is an undetermined game using the definition due to Berner and Juhász.) Finally, we show that various variations on the game yield equivalent games.
10
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
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.
11
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We give several topological/combinatorial conditions that, for a filter on ω, are equivalent to being a non-meager 𝖯-filter. In particular, we show that a filter is countable dense homogeneous if and only if it is a non-meager 𝖯-filter. Here, we identify a filter with a subspace of $2^{ω}$ through characteristic functions. Along the way, we generalize to non-meager 𝖯-filters a result of Miller (1984) about 𝖯-points, and we employ and give a new proof of results of Marciszewski (1998). We also employ a theorem of Hernández-Gutiérrez and Hrušák (2013), and answer two questions that they posed. Our result also resolves several issues raised by Medini and Milovich (2012), and proves false one "theorem" of theirs. Furthermore, we show that the statement "Every non-meager filter contains a non-meager 𝖯-subfilter" is independent of 𝖹𝖥𝖢 (more precisely, it is a consequence of 𝔲 < 𝔤 and its negation is a consequence of ⋄). It follows from results of Hrušák and van Mill (2014) that, under 𝔲 < 𝔤, a filter has less than 𝔠 types of countable dense subsets if and only if it is a non-meager 𝖯-filter. In particular, under 𝔲 < 𝔤, there exists an ultrafilter with 𝔠 types of countable dense subsets. We also show that such an ultrafilter exists under 𝖬 𝖠(countable).
12
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW