Generalizations of continuity of maps and homeomorphisms for studying 2D digital topological spaces and their applications

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• 作者：Sang-Eon Han¹ ; ^{sehan@jbnu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：54A10 ; secondary ; 54C05 ; 54C08 ; 54F65 ; 68U05 ; 68U10
• 刊名：Topology and its Applications
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：196
• 期：part_PB
• 页码：468-482
• 全文大小：485 K

文摘

The present paper establishes two new maps such as an M-map and an M-isomorphism which are generalizations of a Marcus Wyse (for brevity, M-) continuous map and an M-homeomorphism because an M-continuous map is so rigid that some geometric transformations are not M-continuous maps (see Remark 3.2). Furthermore, it proves that in Z²${\mathbf{Z}}^2$ an M-map and an M  -isomorphism are equivalent to a (digitally) 4-continuous map and a (digitally) 4-isomorphism, respectively. Besides, the paper proves that ${\mathit{SC}}_{\mathit{MA}}^{l_1}$ is M  -isomorphic to ${\mathit{SC}}_{\mathit{MA}}^{l_2}$ if and only if l₁=l₂$l_1=l_2$, where ${\mathit{SC}}_{\mathit{MA}}^l$ means a simple closed Marcus Wyse adjacent (for brevity, MA-) curve with l   elements in Z²${\mathbf{Z}}^2$. Finally, the paper proves that MAC   is equivalent to fe4b15afe823cd786e" title="Click to view the MathML source">DTC(4)$\mathit{DTC}(4)$ (see Theorem 6.7), where MAC is the category whose objects are M  -topological spaces (X,γ_X)$(X,{\gamma }_X)$ with MA-adjacency and morphisms are all M  -maps f:(X,γ_X)→(Y,γ_Y)$f:(X,{\gamma }_X)\to (Y,{\gamma }_Y)$ for every ordered pair of objects (X,γ_X)$(X,{\gamma }_X)$ and fe2a85e665d25deaf8e731" title="Click to view the MathML source">(Y,γ_Y)$(Y,{\gamma }_Y)$, and fe4b15afe823cd786e" title="Click to view the MathML source">DTC(4)$\mathit{DTC}(4)$ is the category whose objects are digital images (X,k)$(X,k)$ in Z²${\mathbf{Z}}^2$ and morphisms are (digitally) 4-continuous maps. Besides, we propose the notion of an MA-retract for compressing 2D digital spaces. Using this new approach, we can substantially study and classify 2D digital topological spaces and 2D digital images.