Remarks on Fixed Point Assertions in Digital Topology

Remarks on Fixed Point Assertions in Digital Topology, 3

We continue the work of [5] and [3], in which are considered papers in the literature that discuss fixed point assertions in digital topology. We discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality under "usual" conditions; or that we improve upon.

数字拓扑的基本概念（数字 $k$-覆盖空间与伪 $k$-覆盖空间）

Essential concepts of digital topology (digital $k$-covering spaces and pseudo $k$-covering spaces)

本文纠正了 Boxer 等人关于数字覆盖空间与伪覆盖空间的错误断言和重复内容，并改进相关缺陷，厘清其等价性等基本概念。

Remarks on pointed digital homotopy

We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eve…

关于数字拓扑$k$-群结构的注记及$AP_1$-$k$-群与$AP_1^\text{*}$-$k$-群的构建

Remarks on the digital-topological $k$-group structures and the development of the $AP_1$-$k$- and $AP_1^\ast$-$k$-group

本文针对数字拓扑群结构的现有两类构建方法，指出$NP$型邻接性的缺陷，提出$AP_u$及$AP_u^*$邻接性，定义$AP_1$-$k$-群与$AP_1^*$-$k$-群，并证明后者与Han氏$DT$-$k$-群等价。