scholarly journals An extension problem of a connectedness preserving map between Khalimsky spaces

Filomat ◽  
2016 ◽  
Vol 30 (1) ◽  
pp. 15-28 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Computer Science ◽  
Mathematical Morphology ◽  
Digital Images ◽  
Digital Topology ◽  
Extension Problem ◽  
Topological Spaces ◽  
Digital Geometry ◽  
Applied Topology ◽  
Continuous Map ◽  
Connected Sets

The goal of the present paper is to study an extension problem of a connected preserving (for short, CP-) map between Khalimsky (K-for brevity, if there is no ambiguity) spaces. As a generalization of a K-continuous map, for K-topological spaces the recent paper [13] develops a function sending connected sets to connected ones (for brevity, an A-map: see Definition 3.1 in the present paper). Since this map plays an important role in applied topology including digital topology, digital geometry and mathematical morphology, the present paper studies an extension problem of a CP-map in terms of both an A-retract and an A-isomorphism (see Example 5.2). Since K-topological spaces have been often used for studying digital images, this extension problem can contribute to a certain areas of computer science and mathematical morphology.

scholarly journals Topological graphs based on a new topology on Zn and its applications

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6313-6328 ◽  
Author(s):  
Sang-Eon Han
Keyword(s):  
Extension Problem ◽  
Topological Spaces ◽  
Natural Numbers ◽  
Topological Graph ◽  
Topological Graphs ◽  
Graph Homomorphism ◽  
Continuous Map ◽  
Higher Dimensional ◽  
Khalimsky Topology ◽  
Digital Spaces

Up to now there is no homotopy for Marcus-Wyse (for short M-) topological spaces. In relation to the development of a homotopy for the category of Marcus-Wyse (for short M-) topological spaces on Z2, we need to generalize the M-topology on Z2 to higher dimensional spaces X ? Zn, n ? 3 [18]. Hence the present paper establishes a new topology on Zn; n 2 N, where N is the set of natural numbers. It is called the generalized Marcus-Wyse (for short H-) topology and is denoted by (Zn, n). Besides, we prove that (Z3, 3) induces only 6- or 18-adjacency relations. Namely, (Z3, 3) does not support a 26-adjacency, which is quite different from the Khalimsky topology for 3D digital spaces. After developing an H-adjacency induced by the connectedness of (Zn; n), the present paper establishes topological graphs based on the H-topology, which is called an HA-space, so that we can establish a category of HA-spaces. By using the H-adjacency, we propose an H-topological graph homomorphism (for short HA-map) and an HA-isomorphism. Besides, we prove that an HA-map (resp. an HA-isomorphism) is broader than an H-continuous map (resp. an Hhomeomorphism) and is an H-connectedness preserving map. Finally, after investigating some properties of an HA-isomorphism, we propose both an HA-retract and an extension problem of an HA-map for studying HA-spaces.

scholarly journals Compression of Khalimsky topological spaces

Filomat ◽  
2012 ◽  
Vol 26 (6) ◽  
pp. 1101-1114 ◽  
Author(s):  
Min Kang ◽  
Sang-Eon Han
Keyword(s):  
Computer Science ◽  
Geometric Transformation ◽  
Topological Spaces ◽  
Digital Geometry

Aiming at the study of the compression of Khalimsky topological spaces which is an interesting field in digital geometry and computer science, the present paper develops a new homotopy thinning suitable for the work. Since Khalimsky continuity of maps between Khalimsky topological spaces has some limitations of performing a discrete geometric transformation, the paper uses another continuity (see Definition 3.4) that can support the discrete geometric transformation and a homotopic thinning suitable for studying Khalimsky topological spaces. By using this homotopy, we can develop a new homotopic thinning for compressing the spaces and can write an algorithm for compressing 2D Khalimsky topological spaces.

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

2021 ◽  
Vol 7 (1) ◽  
pp. 1224-1240
Author(s):  
Sang-Eon Han ◽  
◽  
Saeid Jafari ◽  
Jeong Min Kang ◽  
Sik Lee ◽  
...  
Keyword(s):  
Crucial Role ◽  
Dimensional Space ◽  
Digital Topology ◽  
Topological Spaces ◽  
Khalimsky Topology

<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>

Spaces on which every Continuous Map into a given Space is Constant

1986 ◽  
Vol 38 (6) ◽  
pp. 1281-1298 ◽  
Author(s):  
S. Iliadis ◽  
V. Tzannes
Keyword(s):  
Topological Spaces ◽  
Real Numbers ◽  
Continuous Map ◽  
Countable Space ◽  
Continuous Maps ◽  
Usual Topology ◽  
The Given ◽  
Connected Spaces

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].

scholarly journals Categories of L-Fuzzy Čech Closure Spaces and L-Fuzzy Co-Topological Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1274
Author(s):  
Irina Perfilieva ◽  
Ahmed A. Ramadan ◽  
Enas H. Elkordy
Keyword(s):  
Computer Science ◽  
Fuzzy Systems ◽  
Topological Spaces ◽  
Fuzzy Relations ◽  
Approximation Spaces ◽  
Galois Correspondence ◽  
Closure Spaces

Recently, fuzzy systems have become one of the hottest topics due to their applications in the area of computer science. Therefore, in this article, we are making efforts to add new useful relationships between the selected L-fuzzy (fuzzifying) systems. In particular, we establish relationships between L-fuzzy (fuzzifying) Čech closure spaces, L-fuzzy (fuzzifying) co-topological spaces and L-fuzzy (fuzzifying) approximation spaces based on reflexive L-fuzzy relations. We also show that there is a Galois correspondence between the categories of these spaces.

scholarly journals Fuzzy soft connected sets in fuzzy soft topological spaces II

2017 ◽  
Vol 25 (2) ◽  
pp. 171-177 ◽  
Author(s):  
A. Kandil ◽  
O.A. El-Tantawy ◽  
S.A. El-Sheikh ◽  
Sawsan S.S. El-Sayed
Keyword(s):  
Topological Spaces ◽  
Connected Sets

scholarly journals Generalized Fuzzy Soft Connected Sets in Generalized Fuzzy Soft Topological Spaces

2018 ◽  
Vol 14 (2) ◽  
pp. 7787-7805
Author(s):  
Mohammed Saleh Malfi ◽  
Fathi Hishem Khedr ◽  
Mohamad Azab Abd Allah
Keyword(s):  
Topological Spaces ◽  
Connected Sets ◽  
Separated Sets ◽  
The Relationship

In this paper we introduce some types of generalized fuzzy soft separated sets and study some of their properties. Next, the notion of connectedness in fuzzy soft topological spaces due to Karata et al, Mahanta et al, and Kandil  et al., extended to generalized fuzzy soft topological spaces. The relationship between these types of connectedness in generalized fuzzy soft topological spaces is investigated with the help of number of counter examples.

scholarly journals Weakly chain separated sets in a topological space

2021 ◽  
Vol 52 ◽  
pp. 5-16
Author(s):  
Nikita Shekutkovski ◽  
Zoran Misajleski ◽  
Aneta Velkoska ◽  
Emin Durmishi
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
Connected Sets ◽  
Connected Set ◽  
Separated Sets ◽  
Better Than

In this paper we introduce the notion of pair of weakly chain separated sets in a topological space. If two sets are chain separated in the topological space, then they are weakly chain separated in the same space. We give an example of weakly chain separated sets in a topological space that are not chain separated in the space. Then we study the properties of these sets. Also we mention the criteria for two kind of topological spaces by using the notion of chain. The topological space is totally separated if and only if any two different singletons (unit subsets) are weakly chain separated in the space, and it is the discrete if and only if any pair of different nonempty subsets are chain separated. Moreover we give a criterion for chain connected set in a topological space by using the notion of weakly chain separateness. This criterion seems to be better than the criterion of chain connectedness by using the notion of pair of chain separated sets. Then we prove the properties of chain connected, and as a consequence of connected sets in a topological space by using the notion of weakly chain separateness.

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