Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Closed graph property and Khalimsky spaces

Mehrnaz Pourattar, Fatemah Ayatollah Zadeh Shirazi, Mohammad Reza Mardanbeigi

TL;DR

The paper characterizes all self--maps on Khalimsky spaces $K^n$ (and their Alexandroff compactification $A(K^n)$) that have closed graphs, showing any such map must be constant with value $(2λ_1,...,2λ_n)$ for some integers $λ_i$ with evenness; on $A(K^n)$ closed-graph maps are either this constant or the constant to ∞. It also establishes the analogous rigidity for Khalimsky circle and sphere, and presents a diagram of map-classes with explicit examples to illuminate the relationships among closed-graph, constant, continuous and quasi--continuous maps in digital topology. These results illuminate the rigidity of closed-graph maps in Khalimsky spaces and clarify the role of closed points defined by even coordinates.

Abstract

In the following text for Khalimsky $n-$dimensional space $\mathcal{K}^n$ we show self--map $f:\mathcal{K}^n\to\mathcal{K}^n$ has closed graph if and only if there exist integers $λ_1,\ldots,λ_n$ such that $f$ is a constant map with value $(2λ_1,\cdots,2λ_n)$. We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.

Closed graph property and Khalimsky spaces

TL;DR

The paper characterizes all self--maps on Khalimsky spaces (and their Alexandroff compactification ) that have closed graphs, showing any such map must be constant with value for some integers with evenness; on closed-graph maps are either this constant or the constant to ∞. It also establishes the analogous rigidity for Khalimsky circle and sphere, and presents a diagram of map-classes with explicit examples to illuminate the relationships among closed-graph, constant, continuous and quasi--continuous maps in digital topology. These results illuminate the rigidity of closed-graph maps in Khalimsky spaces and clarify the role of closed points defined by even coordinates.

Abstract

In the following text for Khalimsky dimensional space we show self--map has closed graph if and only if there exist integers such that is a constant map with value . We also show each self--map on Khalimsky circle and Khalimsky sphere which has closed graph is a constant map. The text is motivated by examples.
Paper Structure (3 sections, 4 theorems)

This paper contains 3 sections, 4 theorems.

Table of Contents

  1. Introduction
  2. when sel--map $f:\mathcal{K}^n\to \mathcal{K}^n$ has closed graph?
  3. A diagram

Key Result

Lemma 2.1

For each $x,y\in\mathcal{K}^n$ there exists finite sequence $x=x_1,x_2,\ldots,x_p=y\in\mathcal{K}^n$ such that for each $i\in\{1,\ldots,p-1\}$ we have $V_{\mathcal{K}^n}(x_i)\cap V_{\mathcal{K}^n}(x_{i+1})\neq\varnothing$.

Theorems & Definitions (13)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.5: main
  • proof
  • Example 3.1
  • Example 3.2
  • ...and 3 more