≈ g(1,2)*-CLOSED AND ≈ g(1,2)*-OPEN MAPS

General topology plays vital role in many fields of applied sciences as well as in all branches of mathematics. In reality it is used in data mining, computational topology for geometric design and molecular design, computer-aided design, computer-aided geometric design, digital topology, information systems, particle physics and quantum physics etc. By researching generalizations of closed sets, some new separation axioms have been founded and they turn out to be useful in the study of digital topology. Therefore, all bi-topological sets and functions defined will have many possibilities of applications in digital topology and computer graphics.

The effect of some of spaces and sets of topology on the types of digital images

There is a growing interest in studying and improving the characteristics of images and objects in the e-commerce environment. Digital topology is concerned with dealing with the properties and features of two-dimensional (2D) or three-dimensional (3D) digital images such as borders, shapes, the intensity of illumination, and other characteristics. This paper aims to introduce and study new classes of fg-disconnected space and compactly fg-closed set, which could impact the brightness and brightness of the internal components of the types of color images, gray and binary. The paper also aims to find the effect of implementing fg-disconnected space and compactly fg-closed set to determine the brightness and brightness of the internal components of the types of color images, gray and binary. Each research plate contains 30 images of each type of image. Ten different images were chosen at the same time to be analyzed and executed using the proposed system based on MATLAB software. The study proved that higher brightness and light will disappear and delete the components of the image of any kind. This aimed to make the image white and opposite color, the greater darkness, and luminescence will make the picture color mysterious and turn to black.

Convexity and freezing sets in digital topology

<p>We continue the study of freezing sets in digital topology, introduced in [4]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that X is convex.</p>

Remarks on topological spaces on $ {\mathbb Z}^n $ which are related to the Khalimsky $ n $-dimensional space

<abstract><p>The present paper intensively studies various properties of certain topologies on the set of integers $ {\mathbb Z} $ (resp. $ {\mathbb Z}^n $) which are either homeomorphic or not homeomorphic to the typical Khalimsky line topology (resp. $ n $-dimensional Khalimsky topology). This finding plays a crucial role in addressing some problems which remain open in the field of digital topology.</p></abstract>

Remarks on fixed point assertions in digital topology, 4

We continue the work of [4, 2, 3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.

Digital Topological Properties of an Alignment of Fixed Point Sets

The present paper investigates digital topological properties of an alignment of fixed point sets which can play an important role in fixed point theory from the viewpoints of computational or digital topology. In digital topology-based fixed point theory, for a digital image ( X , k ) , let F ( X ) be the set of cardinalities of the fixed point sets of all k-continuous self-maps of ( X , k ) (see Definition 4). In this paper we call it an alignment of fixed point sets of ( X , k ) . Then we have the following unsolved problem. How many components are there in F ( X ) up to 2-connectedness? In particular, let C k n , l be a simple closed k-curve with l elements in Z n and X : = C k n , l 1 ∨ C k n , l 2 be a digital wedge of C k n , l 1 and C k n , l 2 in Z n . Then we need to explore both the number of components of F ( X ) up to digital 2-connectivity (see Definition 4) and perfectness of F ( X ) (see Definition 5). The present paper addresses these issues and, furthermore, solves several problems related to the main issues. Indeed, it turns out that the three models C 2 n n , 4 , C 3 n − 1 n , 4 , and C k n , 6 play important roles in studying these topics because the digital fundamental groups of them have strong relationships with alignments of fixed point sets of them. Moreover, we correct some errors stated by Boxer et al. in their recent work and improve them (see Remark 3). This approach can facilitate the studies of pure and applied topologies, digital geometry, mathematical morphology, and image processing and image classification in computer science. The present paper only deals with k-connected spaces in DTC. Moreover, we will mainly deal with a set X such that X ♯ ≥ 2 .

Fixed point sets in digital topology, 2

We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the complement of the fixed point set.