Submaximality on Bigeneralized Topological Spaces

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.

The aim of this paper is to introduce the concepts of regularly continuous and fully continuous mappings as the mappings that have the preimages of regular open sets are regular open (respectively, open ). We investigate the connections between these classes and several well-known others of `generalized continuous mappings'. Several characterizations and decompositions of certain continuities are provided. In particular, a decomposition of complete continuity, which was first introduced by Arya and Gupta [1] , is also provided. Keywords: Regular open set, Preopen set, t-set, Continuous mapping.

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.

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

The aim of the present paper is to introduce and study different forms of continuity in fuzzy bitopological ordered spaces. The concepts of different mappings such as pairwise fuzzy I -continuous mappings, pairwise fuzzy D -continuous mappings, pairwise fuzzy B -continuous mappings, pairwise fuzzy I -open mappings, pairwise fuzzy D -open mappings, pairwise fuzzy B -open mappings, pairwise fuzzy I -closed mappings, pairwise fuzzy D -closed mappings and pairwise fuzzy B -closed mappings have been introduced. Some of the basic properties and characterization theorems of these mappings have been investigated.

Chapter 9 - Concrete Categories

Bifurcation Theory and Periodic Solutions of Some Autonomous Functional Differential Equations

Functorial admissible quasi-uniformities on topological spaces

In this paper, we investigate the preservation of some topological properties during the weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps on topological spaces. In addition, we also investigate the preservation of topological properties during such corresponding homeomorphisms. Furthermore, we also establish some significant relationships among weakly discontinuous, scatteredly continuous, and point-wise discontinuous maps. Scatteredly continuous map, weakly discontinuous map, point-wise discontinuous map, weak homeomorphism, scattered homeomorphism, point-wise homeomorphism.

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.

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.

In this paper we define the concepts of g -Λ δ -continuous maps, g -Λ δ -irresolute maps and g -V δ -closed maps by using generalized Λ δ -sets and generalized V δ -sets. Also we introduce a new class of topological spaces called T V δ -spaces.

Assume that $X$ is a non-empty set and $T$ and $S$ are real or complex mappings defined on the product $X \times X$. Additive and multiplicative Sincov's equations are: $$T(x,z) = T(x, y ) + T(y, z)$$ and $$S(x,z) = S(x, y ) \cdot S(y, z),$$ respectively. Both equations play important roles in many areas of mathematics. In the present paper we study related inequalities. We deal with functional inequality $$ G(x,z) \leq G(x, y ) \cdot G(y, z), \quad x , y, z \in X $$ and we assume that $X$ is a topological space and $G\colon X \times X \to \mathbb{R}$ is a continuous mapping. In some our statements a considerably weaker regularity than continuity of $G$ is needed. We also study the reverse inequality: $$F(x,z) \geq F(x, y ) \cdot F(y, z), \quad x , y, z \in X $$ and the additive inequality (the triangle inequality): $$H(x,z) \leq H(x,y) + H(y,z), \quad x, y , z \in X.$$ A corollary for generalized (non-symmetric) metric is derived.

The concept of fuzzy Fréchet space Z introduced in two equivalent ways, the first way when Z considered as a fuzzy F − locally convex topological vector space and the second one when Z defined as a complete fuzzy topological vector space whose fuzzy topology generated by a countable separating collection of fuzzy seminorms. Moreover, many classes of continuous mappings between these spaces are established and specified that the class of continuous, strongly continuous and sequentially continuous mappings between these spaces are equivalent.

We consider continuous mappings between two Banach spaces that depend on a parameter with values in a topological space. These mappings are assumed to be continuously differentiable for each value of the parameter. Under normality (regularity) assumptions of the mappings under consideration, we obtain sufficient conditions for the existence of global and semilocal implicit functions. A priori estimates for solutions are given. As an application of these results, we obtain, in particular, a theorem on extending an implicit function from a given closed set to the whole parameter space and a theorem on coincidence points of mappings. Bibliography: 32 titles.