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1
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A category Ψ-density topology

100%
Wilczyński W., Wojdowski W.
Open Mathematics
|
2011
|
tom 9
|
nr 5
1057-1066
EN
Ψ-density point of a Lebesgue measurable set was introduced by Taylor in [Taylor S.J., On strengthening the Lebesgue Density Theorem, Fund. Math., 1958, 46, 305–315] and [Taylor S.J., An alternative form of Egoroff’s theorem, Fund. Math., 1960, 48, 169–174] as an answer to a problem posed by Ulam. We present a category analogue of the notion and of the Ψ-density topology. We define a category analogue of the Ψ-density point of the set A at a point x as the Ψ-density point at x of the regular open representation of A.
2
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On derivations of quantales

100%
Xiao Q., Liu W.
Open Mathematics
|
2016
|
tom 14
|
nr 1
338-346
EN
A quantale is a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins. We define the notions of right (left, two) sided derivation and idempotent derivation and investigate the properties of them. It’s well known that quantic nucleus and quantic conucleus play important roles in a quantale. In this paper, the relationships between derivation and quantic nucleus (conucleus) are studied via introducing the concept of pre-derivation.
3
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On the structure of perfect sets in various topologies associated with tree forcings

76%
Nowik A., Reardon P.
Open Mathematics
|
2013
|
tom 11
|
nr 3
509-518
EN
We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.
4
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The characterizations of upper approximation operators based on special coverings

76%
Wang P., Li Q.
Open Mathematics
|
2017
|
tom 15
|
nr 1
193-202
EN
In this paper, we discuss the approximation operators [...] apr¯NS ${\overline {apr} _{NS}}$ and [...] apr¯S ${\overline {apr} _S}$ which are based on NS(U) and S. We not only obtain some properties of NS(U) and S, but also give examples to show some special properties. We also study sufficient and necessary conditions when they become closure operators. In addition, we give general and topological characterizations of the covering for two types of covering-based upper approximation operators being closure operators.
5
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On the ideal (v 0)

64%
Kalemba P., Plewik S., Wojciechowska A.
Open Mathematics
|
2008
|
tom 6
|
nr 2
218-227
EN
The σ-ideal (v 0) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v 0) to the family of Ramsey null sets. To describe add(v 0) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v 0) = add(v 0) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v 0) = ω 1 implies that (v 0) has the ideal type (c, ω 1, c).
6
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On similarity between topologies

52%
Bartoszewicz A., Filipczak M., Kowalski A., Terepeta M.
Open Mathematics
|
2014
|
tom 12
|
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.
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