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1
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On topologies generated by some operators

100%
Flak K., Hejduk J.
Open Mathematics
|
2013
|
tom 11
|
nr 2
349-356
EN
The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.
2
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Topology from Neighbourhoods

100%
Coghetto R.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
289-296
EN
Using Mizar [9], and the formal topological space structure (FMT_Space_Str) [19], we introduce the three U-FMT conditions (U-FMT filter, U-FMT with point and U-FMT local) similar to those VI, VII, VIII and VIV of the proposition 2 in [10]: If to each element x of a set X there corresponds a set B(x) of subsets of X such that the properties VI, VII, VIII and VIV are satisfied, then there is a unique topological structure on X such that, for each x ∈ X, B(x) is the set of neighborhoods of x in this topology. We present a correspondence between a topological space and a space defined with the formal topological space structure with the three U-FMT conditions called the topology from neighbourhoods. For the formalization, we were inspired by the works of Bourbaki [11] and Claude Wagschal [31].
3
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κ-strong sequences and the existence of generalized independent families

76%
Jureczko J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
1277-1282
EN
In this paper we will show some relations between generalized versions of strong sequences introduced by Efimov in 1965 and independent families. We also show some inequalities between cardinal invariants associated with these both notions.
4
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The closures of arithmetic progressions in the common division topology on the set of positive integers

64%
Szczuka P.
Open Mathematics
|
2014
|
tom 12
|
nr 7
1008-1014
EN
In this paper we characterize the closures of arithmetic progressions in the topology T on the set of positive integers with the base consisting of arithmetic progressions {an + b} such that if the prime number p is a factor of a, then it is also a factor of b. The topology T is called the common division topology.
5
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Boolean algebras admitting a countable minimally acting group

64%
Błaszczyk A., Kucharski A., Turek S.
Open Mathematics
|
2014
|
tom 12
|
nr 1
46-56
EN
The aim of this paper is to show that every infinite Boolean algebra which admits a countable minimally acting group contains a dense projective subalgebra.
6
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A note on pseudobounded paratopological groups

64%
Lin F., Lin S., Sánchez I.
Topological Algebra and its Applications
|
2014
|
tom 2
|
nr 1
EN
Let G be a paratopological group. Then G is said to be pseudobounded (resp. ω-pseudobounded) if for every neighbourhood V of the identity e in G, there exists a natural number n such that G = Vn (resp.we have G = ∪ n∈N Vn). We show that every feebly compact (2-pseudocompact) pseudobounded (ω-pseudobounded) premeager paratopological group is a topological group. Also,we prove that if G is a totally ω-pseudobounded paratopological group such that G is a Lusin space, then is G a topological group. We present some examples of paratopological groups with interesting properties: (1) There exists a metrizable, zero-dimensional and pseudobounded topological group; (2) There exists a Hausdorff ω-pseudobounded paratopological group G such that G contains a dense subgroup which is not ω-pseudobounded; (3) There exists a Hausdorff connected paratopological group which is not ω-pseudobounded.
7
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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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