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The paper examines various properties of D T - k -subgroup structures and addresses an open problem on the existence of topological group structures on the n -dimensional Khalimsky (K -, for brevity) topological space and Marcus–Wyse (M -, for short) topological plane. In particular, we obtain many types of totally k -disconnected or k -connected subgroups of a k -connected D T - k -group. Besides, we prove that each of the n -dimensional K -topological space and the M -topological plane cannot be a typical topological group. Unlike an existence of a D T - k -group structure of (S C k n , l , ∗) (see Proposition 4.7), we prove that neither of (S C K n , l , ∗) and (S C M l , ∗) is a topological group, where S C K n , l (respectively, S C M l ) is a simple closed K - (respectively, M -) topological curve with l elements in ℤ n (respectively, ℤ 2 ) and the operation "∗" is a special kind of binary operation for establishing a group structure of each of S C K n , l and S C M l . Finally, given a D T - k -group structure of (S C k 1 n 1 , l 1 × S C k 2 n 2 , l 2 , ∗) , we find several types of D T - k -subgroup structures of it (see Theorem 5.7). [ABSTRACT FROM AUTHOR]

The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ($ MW $-, for brevity) topology. The Alexandroff compactification of the $ MW $-topological plane is called the infinite $ MW $-topological sphere up to homeomorphism. We first prove that under the $ MW $-topology on $ {\mathbb Z}^2 $ the connectedness of $ X(\subset {\mathbb Z}^2) $ with $ X^\sharp\geq 2 $ implies the semi-openness of $ X $. Besides, for the infinite $ MW $-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $ MW $-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $ MW $-topological sphere is a semi-regular space; thus, it is a semi-$ T_3 $-space because it is a semi-$ T_1 $-space. Hence we finally conclude that an Alexandroff compactification of the $ MW $-topological plane preserves the semi-$ T_3 $ separation axiom.

The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ( M W -, for brevity) topology. The Alexandroff compactification of the M W -topological plane is called the infinite M W -topological sphere up to homeomorphism. We first prove that under the M W -topology on Z 2 the connectedness of X (⊂ Z 2) with X ♯ ≥ 2 implies the semi-openness of X. Besides, for the infinite M W -topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the M W -topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite M W -topological sphere is a semi-regular space; thus, it is a semi- T 3 -space because it is a semi- T 1 -space. Hence we finally conclude that an Alexandroff compactification of the M W -topological plane preserves the semi- T 3 separation axiom. [ABSTRACT FROM AUTHOR]

The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $ MW $-topological plane or $ ({\mathbb Z}^2, \gamma) $ for brevity. We first prove that while every simple closed $ MW $-curve is semi-open in $ ({\mathbb Z}^2, \gamma) $, it may not be semi-closed. Given a simple closed $ MW $-curve with $ l $ elements, denoted by $ SC_{\gamma}^l $, after establishing a continuous analog of $ SC_{\gamma}^l $ denoted by $ \mathcal{A}(SC_{\gamma}^l) $, we initially show that $ \mathcal{A}(SC_{\gamma}^l) $ is both semi-open and semi-closed in $ ({\mathbb R}^2, \mathcal{U}) $, where $ ({\mathbb R}^2, \mathcal{U}) $ is the $ 2 $-dimensional real plane $ {\mathbb R}^2 $ with the usual topology $ \mathcal{U} $. Furthermore, we find a condition for $ \mathcal{A}(SC_{\gamma}^l) $ to separate $ ({\mathbb R}^2, \mathcal{U}) $ into exactly two non-empty components, compared to a typical Jordan curve theorem on $ ({\mathbb R}^2, \mathcal{U}) $. Since not every $ SC_{\gamma}^l $ always separates ($ {\mathbb Z}^2, \gamma) $ into two nonempty components, we find a condition for $ SC_{\gamma}^l, l\neq 4, $ to separate $ ({\mathbb Z}^2, \gamma) $ into exactly two components. The semi-Jordan curve theorem on the $ MW $-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.

Since the Marcus-Wyse ($ MW $-, for brevity) topological spaces play important roles in the fields of pure and applied topology (see Remark 2.2), the paper initially proves that the $ MW $-topological space satisfies the semi-$ T_3 $-separation axiom. To do this work more efficiently, we first propose several techniques discriminating between the semi-openness or the semi-closedness of a set in the $ MW $-topological space. Using this approach, we suggest the condition for simple $ MW $-paths to be semi-closed, which confirms that while every $ MW $-path $ P $ with $ \vert\, P\, \vert\geq 2 $ is semi-open, it may not be semi-closed. Besides, for each point $ p \in {\mathbb Z}^2 $ the smallest open neighborhood of the point $ p $ is proved to be a regular open set so that it is semi-closed. Note that the $ MW $-topological space is proved to satisfy the semi-$ T_3 $-separation axiom, i.e., it is proved to be a semi-$ T_3 $-space so that we can confirm that it also satisfies an $ s $-$ T_3 $-separation axiom. Finally, we prove that the semi-$ T_3 $-separation axiom is a semi-topological property.

The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., M W -topological plane or ( Z 2 , γ) for brevity. We first prove that while every simple closed M W -curve is semi-open in ( Z 2 , γ) , it may not be semi-closed. Given a simple closed M W -curve with l elements, denoted by S C γ l , after establishing a continuous analog of S C γ l denoted by A (S C γ l) , we initially show that A (S C γ l) is both semi-open and semi-closed in ( R 2 , U) , where ( R 2 , U) is the 2 -dimensional real plane R 2 with the usual topology U . Furthermore, we find a condition for A (S C γ l) to separate ( R 2 , U) into exactly two non-empty components, compared to a typical Jordan curve theorem on ( R 2 , U). Since not every S C γ l always separates ( Z 2 , γ) into two nonempty components, we find a condition for S C γ l , l ≠ 4 , to separate ( Z 2 , γ) into exactly two components. The semi-Jordan curve theorem on the M W -topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science. [ABSTRACT FROM AUTHOR]

The work on digitizing subspaces of the 2-D Euclidean space with a certain digital approach is an important discipline in both digital geometry and topology. The present work considers Marcus-Wyse topological approach which was established for studying 2-D digital spaces, Z2. We introduce the digital singular homology groups of MA-spaces (M-topological space with an M-adjacency), and we compute singular homology groups of some certain MA-spaces, we give a formula for singular homology groups of 2-D simple closed MA-curves, and an algorithm for determining homology groups of an arbitrary MA-space. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space ( Z 2 , γ ) . This compactification is called the infinite M-topological sphere and denoted by ( ( Z 2 ) ∗ , γ ∗ ) , where ( Z 2 ) ∗ : = Z 2 ∪ { ∗ } , ∗ ∉ Z 2 and γ ∗ is the topology for ( Z 2 ) ∗ induced by the topology γ on Z 2 . With the topological space ( ( Z 2 ) ∗ , γ ∗ ) , since any open set containing the point “ ∗ ” has the cardinality ℵ 0 , we call ( ( Z 2 ) ∗ , γ ∗ ) the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does ( ( Z 2 ) ∗ , γ ∗ ) have the fixed point property (FPP, for short)? The present paper proves that ( ( Z 2 ) ∗ , γ ∗ ) has the FPP in the category M o p ( γ ∗ ) whose object is the only ( ( Z 2 ) ∗ , γ ∗ ) and morphisms are all continuous self-maps g of ( ( Z 2 ) ∗ , γ ∗ ) such that | g ( ( Z 2 ) ∗ ) | = ℵ 0 with ∗ ∈ g ( ( Z 2 ) ∗ ) or g ( ( Z 2 ) ∗ ) is a singleton. Since ( ( Z 2 ) ∗ , γ ∗ ) can be a model for a digital sphere derived from the M-topological space ( Z 2 , γ ) , it can play a crucial role in topology, digital geometry and applied sciences.

The present paper establishes a new homotopy, called an LMA -homotopy, and further, an LMA -homotopy equivalence suitable for studying the homotopic properties of both Euclidean topology and M -topology. Indeed, the LMA -map (see Definition 12 of the present paper) is an advanced version of that of [18] . Besides, the paper studies relations among an ordinary homotopy equivalence ( resp. contractibility) for spaces ( X , U X ) , an LMA -homotopy equivalence ( resp. LMA -contractibility) for spaces ( X , U X ) and an MA -homotopy equivalence ( resp. MA -contractibility) for MA -spaces. Finally, we classify ( X , U X ) in terms of the LMA -homotopy equivalence. This approach will facilitate studies of applied topology, approximation theory and digital geometry. [ABSTRACT FROM AUTHOR]

The present paper studies both the fixed point property ( FPP for brevity) and the almost fixed point property ( AFPP for short) for Marcus-Wyse ( M -, for short) topological spaces by using an M -retract. Under M -topology, let S N M ( x ) be the smallest open neighborhood of x . In this paper we prove that every M -connected subspace of S N M ( x ) has the FPP . Let S C M 4 be a simple closed M -curve with four elements in Z 2 . Besides, consider an M -topological plane ( X , γ X ) (see Definition 5 ) such that S N M ( x ) ⊂ ( X , γ X ) , where x is a double even point or an even point. Then, proving that S C M 4 is an M -retract of ( X , γ X ) , we propose that every M -topological plane does not have the AFPP . Finally, the present paper proves the following: consider M -connected spaces ( X i , γ X i ) , where X i ⊂ Z , 2 ≤ | X i | ≨ ∞ , i ∈ { 1 , 2 } . Then the Cartesian product as an M -topological subspace ( X 1 × X 2 , γ X 1 × X 2 ) does not have the AFPP . [ABSTRACT FROM AUTHOR]

The paper compares the fixed point property (FPP for short) of a compact Euclidean plane with its digital versions associated with Khalimsky and Marcus-Wyse topology. More precisely, by using a Khalimsky and a Marcus-Wyse topological digitization, the paper studies digital versions of the FPP for Euclidean topological spaces. Besides, motivated by the digital homotopy fixed point property (DHFP for brevity) [O. Ege, I. Karaca, C. R. Math. Acad. Sci. Paris, 353 (2015), 1029-1033], the present paper establishes the digital homotopy almost fixed point property (DHAFP for short) which is more generalized than the DHFP. Moreover, the present paper corrects some errors in [O. Ege, I. Karaca, C. R. Math. Acad. Sci. Paris, 353 (2015), 1029-1033] and improves it. [ABSTRACT FROM AUTHOR]

The study of 2D digital spaces plays an important role in both topology and digital geometry. To propose a certain method of digitizing subspaces of the 2D Euclidean space (or Hausdorff space, denoted by R 2 ), the present paper follows a Marcus–Wyse ( M -, for short) topological approach because the M -topology was developed for studying digital spaces in Z 2 , where Z 2 is the set of points in R 2 with integer coordinates. Hence the present paper uses several tools associated with M -topology, e.g. an M -localized neighborhood of a point p ∈ Z 2 , a topological graph (or a connectedness graph) induced by the M -topology (or M -connectedness graph), a new type of lattice-based connectedness graph homomorphism (or lattice-based M -adjacency map, L M A -map for brevity) which are substantially helpful to M A -digitize subspaces of R 2 , where “ M A ” means the M -adjacency (see Definition 10 and Theorem 3.9 of the present paper). Besides, the paper proposes an algorithm supporting an M A -digitization of subspaces of R 2 . Furthermore, to investigate a relation between subspaces of R 2 and their corresponding M A -digitized spaces, and to classify subspaces of R 2 associated with the M -topology, the paper uses both the first homotopy group (or the fundamental group) in algebraic topology and an M A -fundamental group. [ABSTRACT FROM AUTHOR]

The present paper studies almost fixed point property for digital spaces whose structures are induced by Marcus-Wyse (M-, for brevity) topology. In this paper we mainly deal with spaces X which are connected M-topological spaces withM-adjacency (MA-spaces or M-topological graphs for short) whose cardinalities are greater than 1. Let MAC be a category whose objects, denoted by Ob(MAC), are MA-spaces and morphisms are MA-maps between MA-spaces (for more details, see Section 3), and MTC a category of M-topological spaces as Ob(MTC) and M-continuous maps as morphisms of MTC (for more details, see Section 3). We prove that whereas any MA-space does not have the fixed point property (FPP for short) for any MA-maps, a bounded simple MA-path has the almost fixed point property (AFPP for short). Finally, we refer the topological invariant of the FPP for M-topological spaces from the viewpoint of MTC. [ABSTRACT FROM AUTHOR]

The present paper establishes a homotopy based on Marcus–Wyse (for brevity M -) topology, which can contribute to the classification of 2D digital images in an M -topological approach. Since M -topology is considered in the Euclidean 2D space with integer coordinates, it can be used in studying 2D digital images from the viewpoints of both digital topology and digital geometry such as image processing, image analysis, computer graphics, mathematical morphology and so forth. To develop the homotopy, the present paper uses two maps such as an M -continuous map and Marcus–Wyse adjacency (for brevity MA -) map because they have their own features and merits. Besides, using this homotopy, the present paper proposes an MA -homotopy equivalence and further, MA -contractibility which can be also used in classifying digital images. Finally, the paper establishes an MA -homotopic thinning derived from the above homotopy, which can contribute to the compression of 2D digital images. [ABSTRACT FROM AUTHOR]

The present paper is concerned with the Alexandroff one point compactification of the Marcus-Wyse (M-, for brevity) topological space (Z 2 , γ) . This compactification is called the infinite M-topological sphere and denoted by ((Z 2) ∗ , γ ∗) , where (Z 2) ∗ : = Z 2 ∪ { ∗ } , ∗ ∉ Z 2 and γ ∗ is the topology for (Z 2) ∗ induced by the topology γ on Z 2 . With the topological space ((Z 2) ∗ , γ ∗) , since any open set containing the point " ∗ " has the cardinality ℵ 0 , we call ((Z 2) ∗ , γ ∗) the infinite M-topological sphere. Indeed, in the fields of digital or computational topology or applied analysis, there is an unsolved problem as follows: Under what category does ((Z 2) ∗ , γ ∗) have the fixed point property (FPP, for short)? The present paper proves that ((Z 2) ∗ , γ ∗) has the FPP in the category M o p (γ ∗) whose object is the only ((Z 2) ∗ , γ ∗) and morphisms are all continuous self-maps g of ((Z 2) ∗ , γ ∗) such that | g ((Z 2) ∗) | = ℵ 0 with ∗ ∈ g ((Z 2) ∗) or g ((Z 2) ∗) is a singleton. Since ((Z 2) ∗ , γ ∗) can be a model for a digital sphere derived from the M-topological space (Z 2 , γ) , it can play a crucial role in topology, digital geometry and applied sciences. [ABSTRACT FROM AUTHOR]