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Remarks on Fixed Point Assertions in Digital Topology, 9

Laurence Boxer

Abstract

We continue a discussion of published assertions that are incorrect, incorrectly proven, or trivial, in the theory of fixed points in digital topology.

Remarks on Fixed Point Assertions in Digital Topology, 9

Abstract

We continue a discussion of published assertions that are incorrect, incorrectly proven, or trivial, in the theory of fixed points in digital topology.

Paper Structure

This paper contains 36 sections, 18 theorems, 81 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Adjacencies, continuity, fixed point
  4. Digital metric spaces
  5. Han19's contraction maps
  6. Han19's fixed point property (FPP)
  7. Han19's surfaces
  8. Han19's definition of $k$-DC maps
  9. Han19's Proposition 3.6
  10. Han19's (4.1)
  11. Han19's complexity measure
  12. Han19's finiteness requirement
  13. Han19's Lemma 4.2
  14. Han19's Lemma 4.3
  15. Han19's Theorem 4.4
  16. ...and 21 more sections

Key Result

Theorem 2.5

Bx99 A function $f: X \to Y$ between digital images $(X,\kappa)$ and $(Y,\lambda)$ is $(\kappa,\lambda)$-continuous if and only if for every $x,y \in X$, if $x \leftrightarrow_{\kappa} y$ then $f(x) \leftrightarroweq_{\lambda} f(y)$.

Figures (2)

  • Figure 1: A digital simple closed curve $(X,c_2)$ of 7 points - a counterexample to Han's claim that $\ell$ must be even, where $\ell = \#S_k^{n,\ell}$. $X = \{x_i\}_{i=0}^6$ with each vertex of the graph marked by its index; e.g., $x_0=(0,0)$, $x_1=(1,1)$, etc.
  • Figure 2: Left: a copy of Han19's Figure 4(c), incorrectly claimed to show 8-adjacency, i.e., $c_2$ adjacency. Right: a correct showing of $c_2$ adjacency for this set of vertices.

Theorems & Definitions (46)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 3.1
  • Definition 3.2
  • ...and 36 more