$U(k)$- and $L(k)$-homotopic properties of digitizations of $n$D Hausdorff spaces

Authors : Sangeon HAN

Pages : 127-147

View : 27 | Download : 8

Publication Date : 2017-02-01

Article Type : Research Paper

Abstract :For $X\subset R^n$ let $insert ignore into journalissuearticles values(X, E_X^n);$ be the usual topological space induced by the $n$D Euclidean topological space $insert ignore into journalissuearticles values(R^n, E^n);$ . Based on the upper limit insert ignore into journalissuearticles values($U$-, for short); topology insert ignore into journalissuearticles values(resp. the lower limit insert ignore into journalissuearticles values($L$-, for brevity); topology);, after proceeding with a digitization of  $insert ignore into journalissuearticles values(X, E_X^n);$, we obtain a $U$- insert ignore into journalissuearticles values(resp. an $L$-); digitized space denoted by $D_Uinsert ignore into journalissuearticles values(X);$ insert ignore into journalissuearticles values(resp. $D_Linsert ignore into journalissuearticles values(X);$); in $Z^n$ [16]. Further considering  $D_Uinsert ignore into journalissuearticles values(X);$ insert ignore into journalissuearticles values(resp. $D_Linsert ignore into journalissuearticles values(X);$);  with a digital $k$-connectivity, we obtain a digital image from the viewpoint of digital topology in a graph-theoretical approach, i.e. Rosenfeld model [25], denoted by $D_{Uinsert ignore into journalissuearticles values(k);}insert ignore into journalissuearticles values(X);$ insert ignore into journalissuearticles values(resp. $D_{Linsert ignore into journalissuearticles values(k);}insert ignore into journalissuearticles values(X);$ ); in the present paper. Since a Euclidean topological homotopy has some limitations of studying a digitization of  $insert ignore into journalissuearticles values(X, E_X^n);$,  the present paper establishes the so called $Uinsert ignore into journalissuearticles values(k);$-homotopy insert ignore into journalissuearticles values(resp. $Linsert ignore into journalissuearticles values(k);$-homotopy); which can be used to study homotopic properties of both  $insert ignore into journalissuearticles values(X, E_X^n);$ and  $D_{Uinsert ignore into journalissuearticles values(k);}insert ignore into journalissuearticles values(X);$ insert ignore into journalissuearticles values(resp. both  $insert ignore into journalissuearticles values(X, E_X^n);$ and  $D_{Linsert ignore into journalissuearticles values(k);}insert ignore into journalissuearticles values(X);$ );.  The goal of the paper is to study some relationships among an ordinary homotopy equivalence, a $Uinsert ignore into journalissuearticles values(k);$-homotopy equivalence, an $Linsert ignore into journalissuearticles values(k);$-homotopy equivalence and $k$-homotopy equivalence. Finally, we classify  $insert ignore into journalissuearticles values(X, E_X^n);$ in terms of a $Uinsert ignore into journalissuearticles values(k);$-homotopy equivalence and an $Linsert ignore into journalissuearticles values(k);$-homotopy equivalence. This approach can be used to study applied topology, approximation theory and digital geometry. 
Keywords : U k, digitization, L k, digitization, U and L localized neighborhood, U k, and Linsert ignore into journ