Homotopic properties of an MA-digitization of 2D Euclidean spaces

Özçağ

2020

Mathematics

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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 (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 Mop(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.

Section: Preliminariesmentioning

confidence: 99%

The Fixed Point Property of the Infinite M-Sphere

Özçağ

2020

Mathematics

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“…(2) We say that a grid space is a union of some π-connected components with the given relation π instead of just a π-connected component in a digital space, [13].…”

Section: Preliminariesmentioning

confidence: 99%

“…(2) At (Step 1) in the page 169 of the paper [13], the part "satisfying X ⊂ D" related to an M A-digitization of X should be replaced by "surrounding X" instead. Combining a K-localized neighborhood of Definition 3.1 with an A-map, we define a map G : (X,…”

Section: Definition 38 ([11]mentioning

confidence: 99%

Homotopic properties of $KA$-digitizations of $n$-dimensional Euclidean spaces

Hacettepe Journal of Mathematics and Statistics

2020

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For X(⊂ R n ), assume the subspace (X, E n X ) induced by the n-dimensional Euclidean topological space (R n , E n ). Let Z be the set of integers. Khalimsky topology on Z, denoted by (Z, κ), is generated by the set {{2m − 1, 2m, 2m + 1} | m ∈ Z} as a subbase. Besides, Khalimsky topology on Z n , n ∈ N, denoted by (Z n , κ n ), is a product topology induced by (Z, κ). Proceeding with a digitization of (X, E n X ) in terms of the Khalimsky (K-, for short) topology, we obtain a K-digitized space in Z n , denoted by D K (X)(⊂ Z n ), which is a K-topological space. Considering further D K (X) with K-adjacency, we obtain a topological graph related to the K-topology (a KA-space for short) denoted by D KA (X) (see an algorithm in Section 3). Motivated by an A-homotopy between A-maps for KA-spaces, the present paper establishes a new homotopy, called an LA-homotopy, which is suitable for studying homotopic properties of both (X, E n X ) and D KA (X) because a homotopy for Euclidean topological spaces has some limitations of digitizing (X, E n X ). The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces (X, E n X ), an LA-homotopy equivalence for spaces (X, E n X ), and an A-homotopy equivalence for KA-spaces D KA (X). Finally, we classify KA-spaces (resp. (X, E n X )) via an A-homotopy equivalence (resp. an LA-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.

“…Then (X, B) is called a KA-space pair. Motivated by many kinds of homotopy equivalences [16,18,22,23,28,31], let us consider the notions of an A-homotopy relative to a subset B ⊂ X [30], A-contractibility [30] and an A-homotopy equivalence [30,32,33]. Definition 13.…”

Section: Homotopies In the Category Kac And A Certain Conjecture Invomentioning

confidence: 99%

“…Definition 13. [30,33] Let (X, B) and Y be a space pair and a space in Ob(KAC), respectively. Let f , g : X → Y be A-maps.…”

Section: Homotopies In the Category Kac And A Certain Conjecture Invomentioning

confidence: 99%

Links between Contractibility and Fixed Point Property for Khalimsky Topological Spaces

2019

Mathematics

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Given a Khalimsky (for short, K-) topological space X, the present paper examines if there are some relationships between the contractibility of X and the existence of the fixed point property of X. Based on a K-homotopy for K-topological spaces, we firstly prove that a K-homeomorphism preserves a K-homotopy between two K-continuous maps. Thus, we obtain that a K-homeomorphism preserves K-contractibility. Besides, the present paper proves that every simple closed K-curve in the n-dimensional K-topological space, S C K n , l , n ≥ 2 , l ≥ 4 , is not K-contractible. This feature plays an important role in fixed point theory for K-topological spaces. In addition, given a K-topological space X, after developing the notion of K-contractibility relative to each singleton { x } ( ⊂ X ) , we firstly compare it with the concept of K-contractibility of X. Finally, we prove that the K-contractibility does not imply the K-contractibility relative to each singleton { x 0 } ( ⊂ X ) . Furthermore, we deal with certain conjectures involving the (almost) fixed point property in the categories KTC and KAC, where KTC (see Section 3) (resp. KAC (see Section 5)) denotes the category of K-topological (resp. KA-) spaces, KA-) spaces are subgraphs of the connectedness graphs of the K-topology on Z n .