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New soft separation axioms and fixed soft points with respect to total belong and total non-belong relations

Al-shami Tareq M., Tercan Adnan, Mhemdi Abdelwaheb

Demonstratio Mathematica

2021

Vol. 54, nr 1

196--211

In this article, we exploit the relations of total belong and total non-belong to introduce new soft separation axioms with respect to ordinary points, namely tt-soft pre Ti (i = 0, 1, 2, 3, 4) and tt-soft pre-regular spaces. The motivations to use these relations are, first, cancel the constant shape of soft pre-open and pre-closed subsets of soft pre-regular spaces, and second, generalization of existing comparable properties on classical topology. With the help of examples, we show the relationships between them as well as with soft pre Ti (i = 0, 1, 2, 3, 4) and soft pre-regular spaces. Also, we explain the role of soft hyperconnected and extended soft topological spaces in obtaining some interesting results. We characterize a tt-soft pre-regular space and demonstrate that it guarantees the equivalence of tt-soft pre Ti (i = 0, 1, 2). Furthermore, we investigate the behaviors of these soft separation axioms with the concepts of productand sum of soft spaces. Finally, we introduce a concept of pre-fixed soft point and study its main properties.

Ideal convergence generated by double summability methods

Connor J.

Demonstratio Mathematica

2016

Vol. 49, nr 1

26--37

The main result of this note is that if I is an ideal generated by a regular double summability matrix summability method T that is the product of two nonnegative regular matrix methods for single sequences, then I-statistical convergence and convergence in I-density are equivalent. In particular, the method T generates a density μT with the additive property (AP) and hence, the additive property for null sets (APO). The densities used to generate statistical convergence, lacunary statistical convergence, and general de la Vallée-Poussin statistical convergence are generated by these types of double summability methods. If a matrix T generates a density with the additive property then T-statistical convergence, convergence in T-density and strong T-summabilty are equivalent for bounded sequences. An example is given to show that not every regular double summability matrix generates a density with additve property for null sets.

Modes of ideal continuity and the additive property in the Riesz space setting

Boccuto A., Dimitriou X., Papanastassiou N., Wilczyński W.

Journal of Applied Analysis

2014

Vol. 20, nr 1

41--53

In this paper we present some different types of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the (AP)-property for ideals of an abstract set Λ. Finally we pose some open problems.