Homotopic properties of KAKAKA-digitizations of nnn-dimensional Euclidean spaces

Sang-eon HAN

doi:https://doi.org/10.15672/hujms.546983

Makale detayları

Homotopic properties of KAKAKA-digitizations of nnn-dimensional Euclidean spaces

Yazarlar:

Sang-eon HAN

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 1, pp.236-253

Belge Tipi:

Makale

Yayın Tarihi:

2/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000451269

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.546983

Özet:

 

For X(⊂Rn)X(⊂Rn)X (\subset R^n), assume the subspace (X,EnX)(X,EXn)(X, E_X^n) induced by the nnn-dimensional Euclidean topological space (Rn,En)(Rn,En)(R^n, E^n). Let ZZZ be the set of integers. Khalimsky topology on ZZZ, denoted by  (Z,κ)(Z,κ)(Z, \kappa), is generated by the set {{2m−1,2m,2m+1}|m∈Z}{{2m−1,2m,2m+1}|m∈Z}\{\{2m-1, 2m, 2m+1\}\,\vert\, m \in {Z}\} as a subbase. Besides, Khalimsky topology on  Zn,n∈NZn,n∈NZ^n, n \in N, denoted by (Zn,κn)(Zn,κn)(Z^n, \kappa^n), is a product topology induced by (Z,κ)(Z,κ)({Z}, \kappa). Proceeding with a digitization of (X,EnX)(X,EXn)(X, E_X^n) in terms of the Khalimsky (KKK-, for short) topology, we obtain a KKK-digitized space in ZnZn{Z}^n, denoted by DK(X)(⊂ZnDK(X)(⊂ZnD_K(X) (\subset {Z}^n), which is a KKK-topological space. Considering further DK(X)DK(X)D_K(X) with KKK-adjacency, we obtain a topological graph related to the KKK-topology (a KAKAKA-space for short) denoted by DKA(X)DKA(X)D_{KA}(X) (see an algorithm in Section 3). Motivated by an AAA-homotopy between AAA-maps for KAKAKA-spaces,  the present paper establishes a new homotopy, called an LALALA-homotopy, which is suitable for studying homotopic properties of both (X,EnX)(X,EXn)(X, E_X^n) and DKA(X)DKA(X)D_{KA}(X) because a homotopy for Euclidean topological spaces has some limitations of digitizing (X,EnX)(X,EXn)(X, E_X^n). The goal of the paper is to study some relationships among an ordinary homotopy equivalence for spaces (X,EnX)(X,EXn)(X, E_X^n), an LALALA-homotopy equivalence for spaces (X,EnX)(X,EXn)(X, E_X^n), and an AAA-homotopy equivalence for KAKAKA-spaces DKA(X)DKA(X)D_{KA}(X). Finally, we classify  KAKAKA-spaces (resp. (X,EnX))(X,EXn))(X, E_X^n)) via an AAA-homotopy equivalence (resp. an LALALA-homotopy equivalence). This approach can facilitate studies of applied topology, approximation theory and digital geometry.

Makale detayları

Homotopic properties of KAKAKA-digitizations of nnn-dimensional Euclidean spaces

Yazarlar: Sang-eon HAN

Yayıncı: Hacettepe University

Ana Başlıklar: Matematik

Dergi: Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 1, pp.236-253

Belge Tipi: Makale

Yayın Tarihi: 2/2020

Belge Dili: İngilizce

Document ID (Org No.): 2000451269

Sıklık: İki Aylık

eISSN: 2651-477X

pISSN: 2651-477X

DOI: https://doi.org/10.15672/hujms.546983

Sayı

Yazarlar:

Sang-eon HAN

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2020 Vol.49 Issue 1, pp.236-253

Belge Tipi:

Makale

Yayın Tarihi:

2/2020

Belge Dili:

İngilizce

Document ID (Org No.):

2000451269

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.546983