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Marczewski-Burstin Representations of Boolean Algebras Isomorphic to a Power Set

100%
Bartoszewicz A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2005
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tom 53
|
nr 3
239-250
EN
The paper contains some sufficient conditions for Marczewski-Burstin representability of an algebra 𝓐 of sets which is isomorphic to 𝓟(X) for some X. We characterize those algebras of sets which are inner MB-representable and isomorphic to a power set. We consider connections between inner MB-representability and hull property of an algebra isomorphic to 𝓟 (X) and completeness of an associated quotient algebra. An example of an infinite universally MB-representable algebra is given.
2
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Algebraic and topological properties of some sets in ℓ₁

51%
Banakh T., Bartoszewicz A., Głąb S., Szymonik E.
Colloquium Mathematicum
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2012
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tom 129
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nr 1
75-85
EN
For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $∑_{n=1}^{∞} x(n)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: (𝓘) a finite union of closed intervals; (𝓒) homeomorphic to the Cantor set; 𝓜 𝓒 homeomorphic to the set T of subsums of $∑_{n=1}^{∞} b(n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, 𝓒 and 𝓜 𝓒 the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), (𝓒) and (𝓜 𝓒), respectively. We show that ℐ and 𝓒 are strongly 𝔠-algebrable and 𝓜 𝓒 is 𝔠-lineable. We also show that 𝓒 is a dense $𝒢_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while 𝓒 is not.
3
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Multigeometric sequences and Cantorvals

51%
Bartoszewicz A., Filipczak M., Szymonik E.
Open Mathematics
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2014
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tom 12
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nr 7
1000-1007
EN
For a sequence x ∈ l 1\c 00, one can consider the achievement set E(x) of all subsums of series Σn=1∞ x(n). It is known that E(x) has one of the following structures: a finite union of closed intervals, a set homeomorphic to the Cantor set, a set homeomorphic to the set T of subsums of Σn=1∞ x(n) where c(2n − 1) = 3/4n and c(2n) = 2/4n (Cantorval). Based on ideas of Jones and Velleman [Jones R., Achievement sets of sequences, Amer. Math. Monthly, 2011, 118(6), 508–521] and Guthrie and Nymann [Guthrie J.A., Nymann J.E., The topological structure of the set of subsums of an infinite series, Colloq. Math., 1988, 55(2), 323–327] we describe families of sequences which contain, according to our knowledge, all known examples of x with E(x) being Cantorvals.
4
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Erratum to "Algebraic and topological properties of some sets in ℓ₁" (Colloq. Math. 129 (2012), 75-85)

51%
Banakh T., Bartoszewicz A., Głąb S., Szymonik E.
Colloquium Mathematicum
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2014
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tom 135
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nr 2
295-298
5
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On similarity between topologies

51%
Bartoszewicz A., Filipczak M., Kowalski A., Terepeta M.
Open Mathematics
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2014
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tom 12
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nr 4
603-610
EN
Let T 1 and T 2 be topologies defined on the same set X and let us say that (X, T 1) and (X, T 2) are similar if the families of sets which have nonempty interior with respect to T 1 and T 2 coincide. The aim of the paper is to study how similar topologies are related with each other.
6
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On Marczewski-Burstin representable algebras

51%
Balcerzak M., Bartoszewicz A., Koszmider P.
Colloquium Mathematicum
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2004
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tom 99
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nr 1
55-60
EN
We construct algebras of sets which are not MB-representable. The existence of such algebras was previously known under additional set-theoretic assumptions. On the other hand, we prove that every Boolean algebra is isomorphic to an MB-representable algebra of sets.
7
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Kolmogoroff consistency theorem for Gleason measures

44%
Bartoszewicz A.
Colloquium Mathematicum
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1978
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tom 39
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nr 1
141-151
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