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1
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Properties of the class of measure separable compact spaces

100%
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.
2
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Measures on compact HS spaces

100%
Džamonja M., Kunen K.
Fundamenta Mathematicae
|
1993
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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.
3
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A poset hierarchy

100%
Džamonja M., Thompson K.
Open Mathematics
|
2006
|
tom 4
|
nr 2
225-241
EN
This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem.
4
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Non-separable Banach spaces with non-meager Hamel basis

81%
Banakh T., Džamonja M., Halbeisen L.
Studia Mathematica
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2008
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tom 189
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nr 1
27-34
EN
We show that an infinite-dimensional complete linear space X has: ∙ a dense hereditarily Baire Hamel basis if |X| ≤ 𝔠⁺; ∙ a dense non-meager Hamel basis if $|X| = κ^{ω} = 2^{κ}$ for some cardinal κ.
5
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On bases in Banach spaces

71%
Bartoszyński T., Džamonja M., Halbeisen L., Murtinová E., Plichko A.
Studia Mathematica
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2005
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tom 170
|
nr 2
147-171
EN
We investigate various kinds of bases in infinite-dimensional Banach spaces. In particular, we consider the complexity of Hamel bases in separable and non-separable Banach spaces and show that in a separable Banach space a Hamel basis cannot be analytic, whereas there are non-separable Hilbert spaces which have a discrete and closed Hamel basis. Further we investigate the existence of certain complete minimal systems in $ℓ_{∞}$ as well as in separable Banach spaces.
6
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A consistency result on weak reflection

41%
Cummings J., Džamonja M., Shelah S.
Fundamenta Mathematicae
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1995
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tom 148
|
nr 1
91-100
7
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On squares, outside guessing of clubs and I

41%
Džamonja M., Shelah S.
Fundamenta Mathematicae
|
1995
|
tom 148
|
nr 2
165-198
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