Search Results (13)

Involving w-distances and hybrid contractions that combine conditions of the Ćirić type and Samet et al. type, we obtain some general fixed-point results for quasi-metric spaces from which powerful and significant fixed-point theorems on partial metric spaces are deduced as special cases. We present examples showing that our results are real generalizations of those corresponding to the partial metric case and we give an application to the study of recursive equations where the usual Baire partial metric on a domain of words is replaced with a suitable w-distance. Our approach is inspired on the nice fact, stated by Matthews, that every partial metric induces a weighted quasi-metric. Then, we define the notion of a strong w-distance and deduce that every partial metric is a symmetric strong w-distance for its induced weighted quasi-metric space. Full article

The present research introduces a novel concept termed “neutrosophic fuzzy metric space”, which extends the traditional metric space framework by incorporating the notion of neutrosophic fuzzy sets. A thorough investigation of various structural and topological properties within this newly proposed generalization of metric space has been conducted. Additionally, counterparts of well-known theorems such as the Uniform Convergence Theorem and the Baire Category Theorem have been established for this generalized metric space. Through rigorous analysis, a detailed understanding of its fundamental characteristics has been attained, illuminating its potential applications and theoretical significance. Full article

Let X be a Tychonoff topological space, $B_1(X,\mathbb{R})$ be the space of real-valued Baire 1 functions on X and ${\tau }_{UC}$ be the topology of uniform convergence on compacta. The main purpose of this paper is to study cardinal invariants of $(B_1(X,\mathbb{R}),{\tau }_{UC})$. We prove that the following conditions are equivalent: (1) $(B_1(X,\mathbb{R}),{\tau }_{UC})$ is metrizable; (2) $(B_1(X,\mathbb{R}),{\tau }_{UC})$ is completely metrizable; (3) $(B_1(X,\mathbb{R}),{\tau }_{UC})$ is Čech-complete; and (4) X is hemicompact. It is also proven that if X is a separable metric space with a non isolated point, then the topology of uniform convergence on compacta on $B_1(X,\mathbb{R})$ is seen to behave like a metric topology in the sense that the weight, netweight, density, Lindelof number and cellularity are all equal for this topology and they are equal to $\mathfrak{c}= |B_1(X,\mathbb{R})|$. We find further conditions on X under which these cardinal invariants coincide on $B_1(X,\mathbb{R})$. Full article

This article starts with a study of the congruence of soft sets modulo soft ideals. Different types of soft ideals in soft topological spaces are used to introduce new weak classes of soft open sets. Namely, soft open sets modulo soft nowhere dense sets and soft open sets modulo soft sets of the first category. The basic properties and representations of these classes are established. The class of soft open sets modulo the soft nowhere dense sets forms a soft algebra. Elements in this soft algebra are primarily the soft sets whose soft boundaries are soft nowhere dense sets. The class of soft open sets modulo soft sets of the first category, known as soft sets of the Baire property, is a soft $\sigma$-algebra. In this work, we mainly focus on the soft $\sigma$-algebra of soft sets with the Baire property. We show that soft sets with the Baire property can be represented in terms of various natural classes of soft sets in soft topological spaces. In addition, we see that the soft $\sigma$-algebra of soft sets with the Baire property includes the soft Borel $\sigma$-algebra. We further show that soft sets with the Baire property in a certain soft topology are equal to soft Borel sets in the cluster soft topology formed by the original one. Full article

Soft continuity can contribute to the development of digital images and computational topological applications other than the field of soft topology. In this work, we study a new class of generalized soft continuous functions defined on the class of soft open sets modulo soft sets of the first category, which is called soft functions with the Baire property. This class includes all soft continuous functions. More precisely, it contains various classes of weak soft continuous functions. The essential properties and operations of the soft functions with the Baire property are established. It is shown that a soft continuous with values in a soft second countable space is identical to a soft function with the Baire property, apart from a topologically negligible soft set. Then we introduce two more subclasses of soft functions with the Baire property and examine their basic properties. Furthermore, we characterize these subclasses in terms of soft continuous functions. At last, we present a diagram that shows the relationships between the classes of soft functions defined in this work and those that exist in the literature. Full article

In this paper, we study ⊕-sb-metric spaces, which were introduced to generalize the concept of strong b-metric spaces. In particular, we study the properties of the topology induced via an ⊕-sb metric (separation properties, countability axioms, etc.), prove the continuity of the ⊕-sb-metric, establish the metrizability of the ⊕-sb-metric spaces of countable weight, discuss the convergence structure of an ⊕-sb-metric space and prove the Baire category type theorem for such spaces. Most of the results obtained here are new already for strong b-metric spaces, i.e., in the case where an arithmetic sum “+” is taken in the role of ⊕. Full article

In this paper, we study soft sets of the first and second Baire categories. The soft sets of the first Baire category are examined to be small soft sets from the point of view of soft topology, while the soft sets of the second Baire category are examined to be large. The family of soft sets of the first Baire category in a soft topological space forms a soft $\sigma$-ideal. This contributes to the development of the theory of soft ideal topology. The main properties of these classes of soft sets are discussed. The concepts of soft points where soft sets are of the first or second Baire category are introduced. These types of soft points are subclasses of non-cluster and cluster soft sets. Then, various results on the first and second Baire category soft points are obtained. Among others, the set of all soft points at which a soft set is of the second Baire category is soft regular closed. Moreover, we show that there is symmetry between a soft set that is of the first Baire category and a soft set in which each of its soft points is of the first Baire category. This is equivalent to saying that the union of any collection of soft open sets of the first Baire category is again a soft set of the first Baire category. The last assertion can be regarded as a generalized version of one of the fundamental theorems in topology known as the Banach Category Theorem. Furthermore, it is shown that any soft set can be represented as a disjoint soft union of two soft sets, one of the first Baire category and the other not of the first Baire category at each of its soft points. Full article

Let X, Y be metric spaces and ${\mathcal{B}}_1(X,Y)$ be the space of Baire 1 functions from X to Y. The main purpose of this paper is to study compact subsets of ${\mathcal{B}}_1(X,Y)$ equipped with the topology ${\tau }_{UC}$ of uniform convergence on compacta and prove Ascoli-type theorem for locally bounded Baire 1 functions. The key notion in our paper is the notion of equi-Lebesgue family of functions from X to Y. Full article

In this paper, we will focus on three types of functions in a generalized topological space, namely; lower and upper semi-continuous functions, and cliquish functions. We give some results for nowhere dense sets and for second category sets. Further, we discuss the nature of cliquish functions in generalized metric spaces and provide the characterization theorem for cliquish functions in terms of nowhere dense sets. Full article

In our 2014 work with M. Gabour, we introduced a metric space of generalized nonexpansive self-mappings of bounded and closed subsets of a Banach space and studied, using the Baire category approach, the asymptotic behavior of iterates of a generic operator belonging to this class. In the definition of a generalized nonexpansive mapping the norm is replaced by a general function which can be symmetric as a particular case. In this paper, we prove the convergence of infinite products of generalized nonexpansive self-mappings to a common fixed point in a generic setting. Full article

Ferrando and Lüdkovsky proved that for a non-empty set $\mathsf{\Omega }$ and a normed space X, the normed space $c_0(\mathsf{\Omega },X)$ is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms $\left\{{∥.∥}_n\in \mathbb{N}\right\}$ defining its topology, then $c_0(\mathsf{\Omega },X)$ is the metrizable locally convex space over the field $\mathbb{K}$ (of the real or complex numbers) of all functions $f:\mathsf{\Omega }\to X$ such that for each $\epsilon >0$ and $n\in \mathbb{N}$ the set $\left\{\omega \in \mathsf{\Omega }:{∥f\left(\omega \right)∥}_n>\epsilon \right\}$ is finite or empty, with the topology defined by the semi-norms ${∥f∥}_n=sup\left\{{∥f\left(\omega \right)∥}_n:\omega \in \mathsf{\Omega }\right\}$, $n\in \mathbb{N}$. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space $c_0(\mathsf{\Omega },X)$ is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable $c_0(\mathsf{\Omega },X)$ is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented. Full article

In this paper we study a class of symmetric optimization problems which is identified with a space of objective functions, equipped with an appropriate complete metric. Using the Baire category approach, we show the existence of a subset of the space of functions, which is a countable intersection of open and everywhere dense sets, such that for every objective function from this intersection the corresponding symmetric optimization problem possesses a solution. Full article

A new 122-type phase, monoclinic BaIr₂Ge₂ is successfully synthesized by arc melting; X-ray diffraction and scanning electron microscopy are used to purify the phase and determine its crystal structure. BaIr₂Ge₂ adopts a clathrate-like channel framework structure of the monoclinic BaRh₂Si₂-type, with space group P2₁/c. Structural comparisons of clathrate, ThCr₂Si₂, CaBe₂Ge₂, and BaRh₂Si₂ structure types indicate that BaIr₂Ge₂ can be considered as an intermediate between clathrate and layered compounds. Magnetic measurements show it to be diamagnetic and non-superconducting down to 1.8 K. Different from many layered or clathrate compounds, monoclinic BaIr₂Ge₂ displays a metallic resistivity. Electronic structure calculations performed for BaIr₂Ge₂ support its observed structural stability and physical properties. Full article