REGULARLY CONTINUOUS AND FULLY CONTINUOUS MAPPINGS

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.

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

Let π : Y → X \pi : Y\rightarrow X be a continuous surjection between compact Hausdorff spaces Y Y and X X which is irreducible in the sense that if F ⊊ Y F\subsetneq Y is closed, then π ( F ) ≠ X \pi (F)\neq X . We exhibit isomorphisms between various Boolean algebras associated to this data: the regular open sets of X X , the regular open sets of Y Y , the regular ideals of C ( X ) C(X) and the regular ideals of C ( Y ) C(Y) . We call X X and Y Y Boolean equivalent if the regular open sets of X X and the regular open sets of Y Y are isomorphic Boolean algebras. We give a characterization of when two compact metrizable spaces are Boolean equivalent; this characterization may be viewed as a topological version of the characterization of standard Borel spaces.

In this article, in a bigeneralized topological space, we introduce a new space namely, (s, v)-bigeneralized submaximal space, and analyze its nature. Also, the characterization theorem for a (s, v)-bigeneralized submaximal space, image and preimage of (s, v) bigeneralized submaximal is a (s, v)-bigeneralized submaximal space under (µ, η)-open, (µ, η)-continuous map, respectively are proved. Further, the relationship between hyperconnected space and submaximal space in a pairwise bigeneralized submaximal space is given.

The aim of this paper is to introduce the concept of fuzzy soft tritopological Space, Where we define it as a space equipped with three fuzzy soft topologies, i.e. triple of three fuzzy soft topologies , and on the same initial universal set and the set of parameters .And investigate some of its fundamental properties.Also we defined some new kinds of fuzzy soft open sets in fuzzy soft tritopological spaces, which are called fuzzy soft -open set, fuzzy soft -pre-open set, fuzzy soft semi-open set, fuzzy soft --open set (fuzzy soft tri--open set) and fuzzy soft *open set.We can consider this work as an introduction of this concept.

The purpose of this paper is to provide a new class of generalized N−preopen sets, namely, regular generalized N−preopen sets which is finer than the class of regular generalized preopen sets and the class of regular generalized open sets. Furthermore, we study the fundamental topological properties and introduce the notion of regular generalized N−precontinuous functions.

In this paper we study a weaker form of the classical concept of Menger property. This property, called weakly Menger, is independent from the Menger property and the almost Menger property. In particular, for Tychonoff spaces weaker Menger spaces are not equivalent to almost Menger spaces. We give some characterizations in terms of regular open sets and almost continuous mappings. We also introduce a weaker form of the star-Menger property and the notion of almost γk -set. Some open questions are given.

The concept of complement function is used to define a fuzzy closed subset of a fuzzy topological space. That is a fuzzy subset l is fuzzy closed if the standard complement 1-l = l¢ is fuzzy open. Here the standard complement is obtained by using the function C: [0, 1]® [0, 1] defined by C (x) = 1-x, for all x I[0, 1]. Several fuzzy topologists used this type of complement while extending the concepts in general topological spaces to fuzzy topological spaces. But there are other complements in the fuzzy literature. This motivated the second and third authors to introduce the concepts of fuzzy C -closed sets, and fuzzy C - pre closed sets, in fuzzy topological spaces. In this paper, we generalize the concept of fuzzy strongly pre open and fuzzy strongly pre closed sets by using the arbitrary complement function C, instead of the usual fuzzy complement function, and by using fuzzy C – pre closure instead of fuzzy pre closure. The concepts of fuzzy strongly C – pre open and strongly C – pre closed sets in fuzzy topological spaces and studied their properties in the third and fourth section, respectively. Key words: Fuzzy strongly C – pre open, fuzzy strongly C – pre closed sets and fuzzy topology.

WEAKLY (1,2)$^{\ast}$-$\widetilde{g}$-CLOSED SETS IN BIOTOPOLOGICAL SPACES

One of our first interests in the study of k-spaces (spaces with the weak (= finest) topology generated by their compact subsets) was in the determination of those subsets of a topological space which are themselves k-spaces. If a subset A of a space X is a k-space, thein a subset of A is closed in A if it intersects every compact subset K of X in a set closed in AQnK. If a subset A which is not necessarily a k-space has this property relative to the compact subsets of X, then we say that A has property (k). Throughout this paper, unless exception is noted, we shall assume that compact subsets are closed. A subspace A of a space X is a k-space if and only if A has property (k) and A nK is a k-space for each compact subset K of X (Theorem 1). It follows that an open subset of a k-space in which compact subsets are is itself a k-space. The case when the k-space is Hausdorff was first established independently in [6] by a different proof. This result is similar to a theorem of N. E. Steenrod [7] (an open subset of a k-space is a k-space if it is a regular open set). The class of subspaces of a space X having property (k) is of interest in itself. For example, every subset of X has property (k) if and only if for any set A CX and any point xEA, there exists a compact set K such that xCAnK (Theorem 2). A. Arhangel'skil [1] calls spaces with this property k'-spaces. Analogous to this theorem is a result proved by Arhangel'skil [2] which states that a space X is an hereditary k-space if and only if X is a Frechet-Urysohn space.

L'équation 2i∂∂ƒ = β pour les tenseurs de type (1,1)

: In [7, 9, 13, 14], the class of simply–open sets was introduced and explored. In this paper, we introduce what we call SM− continuity and SMM−continuity and we give several characterizations and two decompositions of SM−continuity. Finally, new decompositions of continuity are provided.

In this paper, we introduce and study the notion of NncZα -open sets and some properties of this class of sets are investigated. Also, we introduce the class of NncA*L sets via NncZα -open sets. Further, by using these sets, a new decompositions of continuous functions are presented.

In this paper, we define new classes of sets called preopen sets, θI-semi-open sets, θI-β-open sets and θI-α-open sets in ideal topological spaces. Also, by using these sets, we obtain new decompositions of continuity in ideal topological spaces.

In this paper, we introduced a new class of sets called γ*-generalized-pre-regular-closed (open) sets and its properties are studied. Also we obtained the notions of γ*-pre-regular- spaces and investigated some of its basic properties. In addition we defined γ*-pre-D sets and γ*-pre-Di spaces (i = 0, 1, 2) using γ*-pre-open sets and discussed their properties comprehensively analyzed their relationship.