Class Of Open Sets Research Articles

Semi-delta-open sets in topological space

This purpose of this paper is to introduce a new class of open sets namely semi-delta-open sets . Further, some basic topological concepts such as neighbourhood axioms, border point, exterior point, frontier point of a set are defined and their properties have been investigated. In addition, in terms of these open sets, semi-delta-closed functions and semi-delta-continuous functions are also defined and their properties have been discussed

A Novel Class of Separation Axioms, Compactness, and Continuity via C-Open Sets

In this paper, we originate a new class of open sets, namely C-open sets, and we review its important properties. Then, some separation axioms of C-open sets are introduced and investigated. In addition, we define the so-called C-compact and C′-compact spaces via C-open sets, and the theorems based on them are discussed with counterexamples. Moreover, we entitle the C-continuous and C′-continuous functions by applying C-open sets. In particular, several inferred properties of them and their connection with the other topological spaces are studied theoretically. Many examples are given to explain the concepts lucidly. The results established in this research paper are new in the field of topology.

Nigh-Open Sets in Topological Space

In this paper, we aim to introduce a new class of open sets namely nigh-open set. Accordingly, we define a topological space called a nigh-topological space. This consequently leads us to outline several new operations in connection with the sets in nigh-topological space coupled with deriving several their properties and relations.

Generalized Investigated in Topological Space

Abstract: In this paper, some characterizations and proportion of notion a investigated. Throughout this paper (X, τ) and (Y, σ ) (simply, X and Y) represent topological spaces on which separation axioms are assumed unless otherwise mentioned. We introduce a new class of sets called regular generalized open sets which is properly placed in between the class of open sets and the class of - open sets. Throughout this paper (X, ) represents a topological space on which no separation axiom is assumed unless otherwise mentioned. For a subset A of a topological space X, cl (A) and int (A) denote the closure of A and the interior of A respectively. X/A or Ac denotes the complement of A in X. introduced and investigated semi open sets, generalized closed sets, regular semi open sets, weakly closed sets, semi generalized closed sets , weakly generalized closed sets, strongly generalized closed sets, generalized pre - regular closed sets, regular generalized closed sets, and generalized -generalized closed sets respectively.

Tilde-gamma-open sets and (Tilde-gamma , Tilde-beta)-continuous mappings

In this paper, we introduce a new class of open sets namely tilde-gamma-open sets in a topological space. In addition, we define tilde-gamma-Ti (i = 0; 1/2; 1; 2) spaces, (tilde-gamma , tilde-beta)-continuous mapping and study their basic properties.

Double Intuitionistic Open Sets with Some Applications

The purpose of this work is to presents a new class of open sets namely Double intuitionistic open sets. The relationships between the Double intuitionistic open and the Double intuitionistic sets are studied including the Double intuitionistic interior set, Double intuitionistic closure set and Double intuitionistic limit point in Double intuitionistic topological spaces were presented and various examples and many observations were presented for each concept, also the definitions of the paired set were presented in general topological spaces Therefore, we generalize it to the Double intuitionistic topological spaces with the presentation of the basic theorem of this new space, and many of the basic characteristics and properties related to these concepts that are presented in the third section as evidence with many examples, taking into account that the opposite is not true for each evidence for these characteristics, which was presented and what we note in the fourth section.

Some Results via Gril Semi –p-Open Set

The significance of the work is to introduce the new class of open sets, which is said Ǥ- -open set with some of properties. Then clarify how to calculate the boundary area for these sets using the upper and lower approximation and obtain the best accuracy.

Scope for application of Topological spaces in Data Granulation through a new class of nearly open set Semi* Regular*- Open sets

The purpose of this paper is to define and study a new class of weaker form of regular*-open sets called semi*- regular*- open sets in topological spaces. Finally we conclude with the scope of this new class of open sets in applications of Data granulation.

αg_Ị-open sets and αg_Ị-functions

The objective of this paper is to show modern class of open sets which is an -open. Some functions via this concept were studied and the relationships such as continuous function strongly -continuous function -irresolute function -continuous function.

Some Inequalities Involving Perimeter and Torsional Rigidity

We consider shape functionals of the form F_q(Omega )=P(Omega )T^q(Omega ) on the class of open sets of prescribed Lebesgue measure. Here q>0 is fixed, P(Omega ) denotes the perimeter of Omega and T(Omega ) is the torsional rigidity of Omega . The minimization and maximization of F_q(Omega ) is considered on various classes of admissible domains Omega : in the class mathcal {A}_{all} of all domains, in the class mathcal {A}_{convex} of convex domains, and in the class mathcal {A}_{thin} of thin domains.

Existence and regularity of Faber-Krahn minimizers in a Riemannian manifold

In this paper, we study the minimization of λ1(Ω), the first Dirichlet eigenvalue of the Laplace-Beltrami operator, within the class of open sets Ω of fixed volume in a Riemannian manifold (M,g). In the Euclidean setting (when (M,g)=(Rn,e)), the well-known Faber-Krahn inequality asserts that the solution of such problem is any ball of suitable volume. Even if similar results are known or may be expected for Riemannian manifolds with symmetries, we cannot expect to find explicit solutions for general manifolds (M,g). In this paper we study existence and regularity properties for this spectral shape optimization problem in a Riemannian setting, in a similar fashion as for the isoperimetric problem. We first give an existence result in the context of compact Riemannian manifolds, and we discuss the case of non-compact manifolds by giving a counter-example to existence. We then focus on the regularity theory for this problem, and using the tools coming from the theory of free boundary problems, we show that solutions are smooth up to a possible residual set of co-dimension 5 or higher.

Theta - ω - Mappings in Topological Spaces

In 2017 S. Ghour and B. Irshedat defined the θω - closure operator as a new topological operator and introduced θω - open sets as a new class of sets and proved that this class of sets is strictly between the class of open sets and the class of θ - open sets. In this paper we introduce θω - continuous, θω - irresolute, θω - open, θω - closed, pre - θω - open, θω - pre closed, contra θω - continuous and almost contra θω - continuous mappings and investigate properties and characterizations of these new types of mappings in topological spaces.

η-Open Sets in Topological Spaces

In this paper a new class of open sets in a topological space called η-open sets is defined and the inclusive relationship of this set with the other existing sets like semi open, pre-open, α-open, r-open sets are discussed.In addition to this the concept of η-interior, η-closure, η-boundary, η-exterior, η-derived, η-border, η-neighbourhood, η-dense, η-residual are also introduced.

The topology of θω-open sets

We define the ??-closure operator as a new topological operator. We show that ??-closure of a subset of a topological space is strictly between its usual closure and its ?-closure. Moreover, we give several sufficient conditions for the equivalence between ??-closure and usual closure operators, and between ??-closure and ?-closure operators. Also, we use the ??-closure operator to introduce ??-open sets as a new class of sets and we prove that this class of sets lies strictly between the class of open sets and the class of ?-open sets. We investigate ??-open sets, in particular, we obtain a product theorem and several mapping theorems. Moreover, we introduce ?-T2 as a new separation axiom by utilizing ?-open sets, we prove that the class of !-T2 is strictly between the class of T2 topological spaces and the class of T1 topological spaces. We study relationship between ?-T2 and ?-regularity. As main results of this paper, we give a characterization of ?-T2 via ??-closure and we give characterizations of ?-regularity via ??-closure and via ??-open sets.

On optimal location of diffusion and related optimal control for null controllable heat equation

We consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi-dimensional heat equations. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa. We formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems. One is seeking the optimal domain of heat diffusion with a fixed interior actuator domain. This can be considered as a domain perturbation problem in shape optimization. Another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain. The existences of these two types of shape optimization problems over some class of open sets in general RN space have been proved separately.

On a Class of γ*-pre-open Sets in Topological Spaces

Abstract. In this paper, a new class of open sets, namely -pre-open sets was intro-duced and its basic properties were studied. Moreover a new type of topology ˝ p wasgenerated using -pre-open sets and characterized the resultant topological space (X,˝ p ) as -pre-T 12 space. 1. IntroductionThe concepts of pre-open sets and semi-pre-open sets were introduced respec-tively by Mashhour et al.[6] and Andrijevic[1]. Andrijevic[1] introduced a new classof topology generated by pre-open sets and corresponding closure and interior op-erators. Kasahara[3] de ned the concept of an operation on topological spaces andintroduced the concept of -closed graphs of an operation. Ogata[7] called the op-eration (respectively -closed set) as -operation (respectively -closed set) andintroduced the notion of ˝ which is the collection of -open sets in a topologicalspace. Further, he de ned the concept of -closure and ˝-closure operators and * Corresponding Author.Received April 6, 2011; accepted September 11, 2012.2010 Mathematics Subject Classi cation: 54A05, 54A10, 54D10.Key words and phrases: -closed(open), -closure, -interior, -pre-closed(open), -pre-closure, -pre-interior, -semi-pre-closed(open), -semi-pre-closure, -semi-pre-interior, ˝-p-closed(open), ˝-p-closure, ˝-p-interior.

Optimal actuator location for time and norm optimal control of null controllable heat equation

We consider minimal time and minimal norm of the optimal controls for a multi-dimensional internally controlled heat equation with control domain varying over a class of open sets. Two problems are formulated separately into different types of shape optimization problems over this open set class. The governing equation with any given initial value and admissible control are considered as constraints, and minimal time or minimal norm of the optimal controls is considered as cost for related shape optimization. The solution of the shape optimization leads to the optimal actuator location for optimal controls. The existence of such an optimal location domain for both minimal time and minimal norm controls is presented. A different problem that relates the balance between minimal time and minimal norm controls is also discussed. This study builds a link between optimal control and shape optimization for this multi-dimensional heat equation.

Some Compact Classes of Open Sets under Hausdorff Distance and Application to Shape Optimization

In this paper, we introduce three new classes of open sets in a general Euclidean space $\mathbb{R}^N$. It is shown that every class of open sets is compact under the Hausdorff distance. The result is then applied to a shape optimization problem of elliptic equation. The existence of the optimal solution is presented.

Quadratic differentials in problems of the product of inner radii of nonoverlapping domains

We study applications of the theory of quadratic differentials to the solution of new extremal problems of nonoverlapping domains with free poles on rays and generalize known results to a certain class of open sets.

On some characterizations of vividly and blurry (l,2)-β-irresolute mappings

The aim of this paper is to introduce and characterize the vividly (1, 2)-?-irresolute mapping and blurry (1, 2)-?-irresolute mapping. We also define (1, 2)-?-T2 spaces and (1, 2)-semi-preregular spaces. These spaces are characterized by a new class of open sets, called (1,2)-semipre-?-open sets. .