The autohomeomorphism group of some Digital Lines

Abstract

For a topological space $X$, the group of autohomeomorphisms is denoted by $\mathcal{H}(X)$. It is a well-established fact that even if two topological spaces $X$ and $Y$ have isomorphic autohomeomorphism groups, it does not necessarily imply that $X$ and $Y$ are homeomorphic. A space $X$ is considered homogeneous if its autohomeomorphism group, $\mathcal{H}(X)$, acts transitively on $X$, via the action \[\begin{array}{ccc}
\mathcal{H}(X)\times X& \longrightarrow& X \\
(g,x) &\longmapsto& g(x).
\end{array}\]

The degree of homogeneity of $X$, denoted as $d_h(X)$, is defined as the cardinality of the quotient set $X/\mathcal{H}(X)$ relative to the aforementioned action.

Regarding the Khalimsky topology defined on the set of integers, this topology, denoted by $\mathcal{K}$, is the topology generated by the family \[\left\{\{x-1,x,x+1\}\colon x\textrm{ is an even integer}\right\}.\]
The space $(\mathbb{Z}, \mathcal{K})$, known as the Khalimsky line or digital line, is called the \emph{Khalimsky line (or digital line)}, will be denote it by $\mathbf{KL}$ (or $\mathbf{KL}_1$). The digital line is notably influential in digital image processing and computer graphics.

The aim of this paper is the construction of a sequence of Alexandroff topologies, $\{\mathcal{K}_p: p\in \mathbf{N}\}$, on the set of integers $\mathbb{Z}$. This provides new digital lines with the following properties:
\begin{itemize}
\item[-] $\mathcal{H}(\mathbb{Z}, \mathcal{K}_p)$ is isomorphic to $\mathcal{H}(\mathbb{Z}, \mathcal{K})$.
\item[-] For each positive integer $p$, $(\mathbb{Z}, \mathcal{K}_p)$ is topologically embedded in $(\mathbb{Z}, \mathcal{K}_{p+1})$.
\item[-] $d_H(\mathbb{Z}, \mathcal{K}_p)=p+1$.
\end{itemize}