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1

Two new forms of ordered soft separation axioms

Al-shami Tareq M., El-Shafei Mohammed E.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

8--26

EN

The goal of this work is to introduce and study two new types of ordered soft separation axioms, namely soft Ti-ordered and strong soft Ti-ordered spaces (i = 0, 1, 2, 3, 4). These two types are formulated with respect to the ordinary points and the distinction between them is attributed to the nature of the monotone neighborhoods. We provide several examples to elucidate the relationships among these concepts and to show the relationships associate them with their parametric topological ordered spaces and p-soft Ti-ordered spaces. Some open problems on the relationships between strong soft Ti-ordered and soft Ti-ordered spaces (i = 2, 3, 4) are posed. Also, we prove some significant results which associate both types of the introduced ordered axioms with some notions such as finite product soft spaces, soft topological and soft hereditary properties. Furthermore, we describe the shape of increasing (decreasing) soft closed and open subsets of soft regularly ordered spaces; and demonstrate that a condition of strong soft regularly ordered is sufficient for the equivalence between p-soft T1-ordered and strong soft T1-ordered spaces. Finally, we establish a number of findings that associate soft compactness with some ordered soft separation axioms initiated in this work.

2

On soft pc-separation axioms

Hamko Qumri H., Ahmed Nehmat K., Khalaf Alias B.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

67--79

EN

Many mathematicians defined and studied soft separation axioms and soft continuity in soft spaces by using ordinary points of a topological space X. Also, some of them studied the same concepts by using soft points. In this paper, we introduce the concepts of soft pc−Ti and soft pc−T⁎i, i=0,1,2 by using the concept of soft pc-open sets in soft topological spaces. We explore several properties of such spaces. We also investigate the relationship among these spaces and provide a counter example when it is needed.

3

Applications of some operators on supra topological spaces

Asaad Baravan A., Al-shami Tareq M., Abo-Tabl El-Sayed A.

Demonstratio Mathematica

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2020

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Vol. 53, nr 1

292--308

EN

In this paper, the notion of an operator γ on a supra topological space(X, μ) is studied and then utilized to analyze supra γ-open sets. The notions of μγ-g. closed sets on the subspace are introduced and investigated. Furthermore, some new μγ-separation axioms are formulated and the relationships between them are shown. Moreover, some characterizations of the new functions via operator γ on μ are presented and investigated. Finally, we give some properties of S(γ, β)-closed graph and strongly S(γ, β)-closed graph.

4

Almost continuity, regular set-connected mappings and some separation axioms

Duszyński Z.

Demonstratio Mathematica

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2006

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Vol. 39, nr 4

939-948

EN

Let f : (X, r ) approaches (Y,sigma) be a mapping, let (X, rs) denote the topological space generated by the family of all regular open subsets of (X, r ) and let fxs : (X, rs) !approaches (Y, sigma) be defined by fxs (x) = f(x) for each x is an element of X. In the paper relationships between almost continuity of f, almost continuity of fxs and some other types of mappings (r.s.c. mappings in particular) are studied.

5

On density topologies with respect to invariant σ-ideals

Hejduk J.

Journal of Applied Analysis

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2002

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Vol. 8, nr 2

201-219

EN

The density topologies with respect to measure and category are motivation to consider the density topologies with respect to invariant σ-ideals on R. The properties of such topologies, including the separation axioms, are studied.