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1
Content available remote More properties of generalized open sets in generalized topological spaces
Sarsak Mohammad S.
Demonstratio Mathematica
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2022
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Vol. 55, nr 1
404--415
EN
Sarsak [M. S. Sarsak, On some properties of generalized open sets in generalized topological spaces, Demonstr. Math. 46 (2013), no. 2, 415–427] studied some properties of generalized open sets in generalized topological spaces (GTSs); the primary purpose of this article is to investigate more properties of generalized open sets in GTSs. We mainly study the behaviours of regular closed sets, semi-open sets, regular semi-open sets, preopen sets, and β -open sets in GTSs analogous to their behaviours in topological spaces.
2
Content available remote More on μ-semi-Lindelöf sets in μ-spaces
Sarsak Mohammad S.
Demonstratio Mathematica
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2021
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Vol. 54, nr 1
259--271
EN
Sarsak [On μ-compact sets in μ-spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49-57] introduced and studied the class of μ-Lindelöf sets in μ-spaces. Mustafa [μ-semi compactness and μ-semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535-541] introduced and studied the class of μ-semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ-semi-Lindelöf sets in μ-spaces. The class of μ-semi-Lindelöf sets in μ-spaces is a proper subclass of the class of μ-Lindelöf sets in μ-spaces. It is shown that the property of being μ-semi-Lindelöf is not monotonic, that is, if (X, μ) is a μ-space and A ⊂ B ⊂ X, where A is μB-semi-Lindelöf, then A need not be μ-semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ωμ-semi-open sets, and investigate them to obtain new properties and characterizations of μ-semi-Lindelöf sets in μ-spaces.
3
Content available remote On some properties of generalized open sets in generalized topological spaces
Sarsak M. S.
Demonstratio Mathematica
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2013
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Vol. 46, nr 2
415--427
EN
In [8], Császár introduced some generalized open sets in generalized topological spaces, namely, µ-semi-open, µ-preopen, µ-β-open, and µ-α-open. The primary purpose of this paper is to study some new properties and characterizations of these concepts as we also introduce and study a new generalized open set in generalized topological spaces, namely, µ-b-open. We also investigate some relationships between these concepts.
4
Content available remote Applications of topology in computer algorithms
Telgarsky R.
Matematyka Stosowana : matematyka dla społeczeństwa
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2011
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T. 13/54
35-46
EN
The aim of this paper is to discuss some applications of general topology in computer algorithms including modeling and simulation, and also in computer graphics and image processing. While the progress in these areas heavily depends on advances in computing hardware, the major intellectual achievements are the algorithms. The applications of general topology in other branches of mathematics are not discussed, since they are not applications of mathematics outside of mathematics.
PL
Celem niniejszego artykułu jest omówienie niektórych zastosowań topologii ogólnej w algorytmach komputerowych w kontekście modelowania i symulacji, a także w grafice komputerowej i przetwarzaniu obrazów. Choć postęp w tych dziedzinach jest bardzo uzależniony od postępów w sprzęcie komputerowym, jednak głównymi osiągnięciami intelektualnymi są algorytmy. Zastosowania topologii ogólnej w innych działach matematyki nie są omawiane, ponieważ nie są one zastosowaniami matematyki poza matematyką.
5
Content available On cluster sets of connected functions
Jędrzejewski J.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2009
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Vol. 14
51-61
EN
In the article we discuss some properties of a few classes of connected functions and cluster sets of functions from those classes.
6
Content available remote Topologies and smooth maps on initial and final objects in the category of Frolicher spaces
Batubenge A., Tshilombo H.
Demonstratio Mathematica
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2009
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Vol. 42, nr 3
641-655
EN
In this paper, we show that when the Frolicher smooth structure is induced on a subset or a quotient set, there are three natural topologies underlying the resulting object. We study these topologies and compare them in each case. It is known that the topology generated by strucure functions is the weakest one in which all functions and curves on the space are continuous. We show that on a subspace, it is rather the trace topology which has this property, while the three topologies are coincident on the quotient space. We construct a base for the Frolicher topology and using either a base or a subbase in the sense of A. Frolicher [9], we characterise the morphisms of this category.
7
Content available remote K-connectivity
Jędrzejewski J., Kowalczyk S.
Scientific Issues of Jan Długosz University in Częstochowa. Mathematics
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2008
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Vol. 13
27-32
EN
The notion of some connectedness is introduced and it is compared with usual connected sets. Subsets of a topological space that are called i-connected have been introduced by J. Knop & M. Wróbel in [2] in 2006 The reason of introducing this notion is based upon the fact that a subset of the space R with nonempty interior is i-connected if and only if it is connected.
8
Content available remote On gβ-regular and gβ-normal spaces
Tahiliani S.
Demonstratio Mathematica
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2007
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Vol. 40, nr 3
739-743
EN
The aim of this paper is to introduce and study two new classes of spaces, called gβ-regular and gβ-normal spaces. The concept of gβ-regularity and gβ-normality are separation properties obtained by utilizing gβ-closed sets. Recall that a subset A of a space (X, r ) is called gβ-closed if β Cl(A) is a subset of U where A is a subset of U and U is open in X.
9
Content available remote A locally connected continuum which is not an IFS attractor
Kwieciński M.
Bulletin of the Polish Academy of Sciences. Mathematics
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1999
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Vol. 47, no 2
127-132
EN
We construct a locally connected continuum embedded in the plane which is not the attractor of any iterated function system.
10
Content available remote Inducible mappings between hyperspaces
Charatonik Janusz J., Charatonik W.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 1
5-9
EN
Given a continuum X we denote by [2^x] and C(X) the hyperspace of all nonempty compact subsets and of all nonempty subcontinua of X. For any two continua X and Y and a mapping [f : X --> Y let 2^f] and C(f) stand for the induced mappings between corresponding hyperspaces. A mapping g between the hypespaces is inducible is there exists a mapping f such that [g = 2^f] or g = C(f), respectively. Necessary and sufficient conditions are shown under which a given mapping g is inducible.
11
Content available remote Dynamical properties of automorphisms of minimal flows
Auslander J., Markley N.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 1
105-112
EN
The dynamics of automorphisms of minimal flows are studied using a group which generalizes the Ellis group of the flow.
12
Content available remote A note on indecomposability of chaotic continua on surfaces
Kato H.
Bulletin of the Polish Academy of Sciences. Mathematics
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1998
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Vol. 46, no 1
11-16
EN
In [5, 6] and [8], we proved that if a plane compactum X with dim X > 0 admits a continuum-wise expansive homeomorphism f, then X is not locally connected and it contains a sigma-chaotic continuum Z (sigma = s or u) of f such that Z is indecomposable and for each z [...] Z, the continuum-wise sigma-stable set V[sup sigma] (z, Z) coincides with the composant containing z. In this note, we generalize this theorem to 1-dimensional compacta in arbitrary surfaces.
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