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1
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Compact scattered spaces in forcing extensions

100%
Kunen K.
Fundamenta Mathematicae
|
2005
|
tom 185
|
nr 3
261-266
EN
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
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The Complex Stone-Weierstrass Property

100%
Kunen K.
Fundamenta Mathematicae
|
2004
|
tom 182
|
nr 2
151-167
EN
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
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The power set of ω Elementary submodels and weakenings of CH

64%
Juhász I., Kunen K.
Fundamenta Mathematicae
|
2001
|
tom 170
|
nr 3
257-265
EN
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
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Properties of the class of measure separable compact spaces

64%
Džamonja M., Kunen K.
Fundamenta Mathematicae
|
1995
|
tom 147
|
nr 3
261-277
EN
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
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Measures on compact HS spaces

64%
Džamonja M., Kunen K.
Fundamenta Mathematicae
|
1993
|
tom 143
|
nr 1
41-54
EN
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
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Measures on Corson compact spaces

64%
Kunen K., van Mill J.
Fundamenta Mathematicae
|
1995
|
tom 147
|
nr 1
61-72
EN
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
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Locally constant functions

64%
Hart J. E., Kunen K.
Fundamenta Mathematicae
|
1996
|
tom 150
|
nr 1
67-96
EN
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
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Bohr compactifications of discrete structures

64%
Hart J. E., Kunen K.
Fundamenta Mathematicae
|
1999
|
tom 160
|
nr 2
101-151
EN
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
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On the open-open game

52%
Daniels P., Kunen K., Zhou H.
Fundamenta Mathematicae
|
1994
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tom 145
|
nr 3
205-220
EN
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
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Hausdorff measures and two point set extensions

52%
Dijkstra J. J., Kunen K., van Mill J.
Fundamenta Mathematicae
|
1998
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tom 157
|
nr 1
43-60
EN
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
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Seven characterizations of non-meager 𝖯-filters

52%
Kunen K., Medini A., Zdomskyy L.
Fundamenta Mathematicae
|
2015
|
tom 231
|
nr 2
189-208
EN
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
Content available remote

On inhomogeneity of products of compact F-spaces

51%
Frankiewicz R., Kunen K., Zbierski P.
Fundamenta Mathematicae
|
1988
|
tom 129
|
nr 1
35-38
13
Content available remote

Solution of Kuratowski's problem on function having the Baire property, I

51%
Frankiewicz R., Kunen K.
Fundamenta Mathematicae
|
1987
|
tom 128
|
nr 3
171-180
14
Content available remote

Between Martin's Axiom and Souslin's Hypothesis

38%
Kunen K., Tall F. D.
Fundamenta Mathematicae
|
1979
|
tom 102
|
nr 3
173-181
15
Content available remote

Maximal σ-independent families

32%
Kunen K.
Fundamenta Mathematicae
|
1983
|
tom 117
|
nr 1
75-80
16
Content available remote

Rigid P-spaces

26%
Kunen K.
Fundamenta Mathematicae
|
1989
|
tom 133
|
nr 1
59-65
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