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Abstract  

Characterizations of some properties of generalized R 0 and R 1 topological spaces by using closure operator defined on a generalized topological space will be given. It is also shown that many results done in this area in some previous papers can be considered as special cases of our results.

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We are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T0, T1, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even they are not really needed).

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Abstract  

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\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{I}_g$$ \end{document}
-regular spaces are introduced and various characterizations and properties are given. Characterizations of normal, mildly normal, g-normal, regular and almost regular spaces are also given.
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Abstract

We introduce and study new separation axioms ingeneralized topological spaces, namely, , and . spaces are strictly placed between μ-T 0 spaces and , spaces are strictly placed between spaces and spaces, and spaces are strictly placed between spaces and μ-T 1 spaces.

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Abstract

A new kind of sets called generalized μ-closed (briefly g μ-closed) sets are introduced and studied in a topological space by using the concept of generalized open sets introduced by Á. Császár. The class of all g μ-closed sets is strictly larger than the class of all μ-closed sets. Furthermore, g-closed sets (in the sense of N. Levine [17]) is a special type of g μ-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of μg-regular and μg-normal spaces have been given.

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Abstract  

For metrically generated constructs X we give an internal characterization of the regular closure operator on X, determined by the subconstruct X 0, consisting of its T 0 objects. This allows us to describe the epimorphisms in X 0 and to show that all the constructs of that type are cowellpowered. We capture many known results but our method also gives solutions in cases where the epimorphism problem was still open.

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Abstract

The main aim of this paper is to show that every GTS can be realized as a μ-closed subspace of a generalized hyperconnected space. Also, we give more characterizations of generalized hyperconnected spaces.

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In Parts II to IV, we are going to investigate simultaneous extensions of various topological structures (i.e. traces on several subsets at the same time are prescribed), also with separation axioms T0, T1, symmetry (in the sense of Part I, § 3), Riesz property, Lodato property. The following questions will be considered: (i) Under what conditions is there an extension? (ii) How can the finest extension be described? (iii) Is there a coarsest extension? (iv) Can we say more about extensions of two structures than in the general case? (v) Assume that certain subfamilies (e.g. the finite ones) can be extended; does the whole family have an extension, too? The general categorial results from Part I will be applied whenever possible (even when they are not really needed).

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Abstract  

We introduce and study the concepts of weak neighborhood systems, weak neighborhood spaces, ω(ψ, ψ′)-continuity, ω-continuity and ω*-continuity on WNS’s.

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Abstract

An explicit characterization of each of the separation properties T i, i=0,1, , and T 2 at a point p is given in the topological category of Cauchy spaces. Moreover, specific relationships that arise among the various T i, i=0,1, , and T 2 structures at p are examined in this category. Finally, we investigate the relationships between generalized separation properties and separation properties at a point p in this category.

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