Let $(X,T)$ be an Alexandroff space. We define the adjacency relation $A{R}_T$ on $X$ induced by $T$ as the irreflexive relation defined for $x\ne y$ in $X$ by:

$$(x,y)\in A{R}_T\mathrm{i}\mathrm{f}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y}\mathrm{i}\mathrm{f}x\in S{N}_T(y)\mathrm{o}\mathrm{r}y\in S{N}_T(x),$$

where $S{N}_T(z)$ is the smallest open set containing $z$ in $(X,T)$ and $z\in \{x,y\}$. Two families of Alexandroff topologies $(T_k,k\in \mathbb{Z})$ and $(T_k^{\mathrm{\prime }},k\in \mathbb{Z})$ have been recently introduced on $\mathbb{Z}$. The aim of this paper is to show that for each nonzero integers $k$, the topologies $T_k,T_k^{\mathrm{\prime }}$, $T_{-k}$, and $T_{-k}^{\mathrm{\prime }}$ are homeomorphic. The adjacency relations induced by the product topologies $(T_k{)}^n$ and $(T_k^{\mathrm{\prime }}{)}^n$ are studied and compared with classical ones. We also show that the adjacency relations induced by $T_k,T_k^{\mathrm{\prime }}$, $T_{-k}$, and $T_{-k}^{\mathrm{\prime }}$ are isomorphic. Then, note that the adjacency relations on $\mathbb{Z}$ induced by these topologies, $k\ne 0$, are different from each other.

Since the Marcus-Wyse ( $MW$-, for brevity) topological spaces play important roles in the fields of pure and applied topology (see Remark 2.2), the paper initially proves that the $MW$-topological space satisfies the semi- $T_3$-separation axiom. To do this work more efficiently, we first propose several techniques discriminating between the semi-openness or the semi-closedness of a set in the $MW$-topological space. Using this approach, we suggest the condition for simple $MW$-paths to be semi-closed, which confirms that while every $MW$-path $P$ with $|P|\ge 2$ is semi-open, it may not be semi-closed. Besides, for each point $p\in {\mathbb{Z}}^2$ the smallest open neighborhood of the point $p$ is proved to be a regular open set so that it is semi-closed. Note that the $MW$-topological space is proved to satisfy the semi- $T_3$-separation axiom, i.e., it is proved to be a semi- $T_3$-space so that we can confirm that it also satisfies an $s$- $T_3$-separation axiom. Finally, we prove that the semi- $T_3$-separation axiom is a semi-topological property.

The present paper aims to investigate some semi-separation axioms relating to the Alexandroff one point compactification (Alexandroff compactification, for short) of the digital plane with the Marcus-Wyse ( $MW$-, for brevity) topology. The Alexandroff compactification of the $MW$-topological plane is called the infinite $MW$-topological sphere up to homeomorphism. We first prove that under the $MW$-topology on ${\mathbb{Z}}^2$ the connectedness of $X(\subset {\mathbb{Z}}^2)$ with $X^{\mathrm{♯}}\ge 2$ implies the semi-openness of $X$. Besides, for the infinite $MW$-topological sphere, we introduce a new condition for the hereditary property of the compactness of it. In addition, we investigate some conditions preserving the semi-openness or semi-closedness of a subset of the $MW$-topological plane in the process of an Alexandroff compactification. Finally, we prove that the infinite $MW$-topological sphere is a semi-regular space; thus, it is a semi- $T_3$-space because it is a semi- $T_1$-space. Hence we finally conclude that an Alexandroff compactification of the $MW$-topological plane preserves the semi- $T_3$ separation axiom.

The paper initially develops the semi-Jordan curve theorem on the digital plane with the Marcus-Wyse topology, i.e., $MW$-topological plane or $({\mathbb{Z}}^2,\gamma )$ for brevity. We first prove that while every simple closed $MW$-curve is semi-open in $({\mathbb{Z}}^2,\gamma )$, it may not be semi-closed. Given a simple closed $MW$-curve with $l$ elements, denoted by $S{C}_{\gamma }^l$, after establishing a continuous analog of $S{C}_{\gamma }^l$ denoted by $\mathcal{A}(S{C}_{\gamma }^l)$, we initially show that $\mathcal{A}(S{C}_{\gamma }^l)$ is both semi-open and semi-closed in $({\mathbb{R}}^2,\mathcal{U})$, where $({\mathbb{R}}^2,\mathcal{U})$ is the $2$-dimensional real plane ${\mathbb{R}}^2$ with the usual topology $\mathcal{U}$. Furthermore, we find a condition for $\mathcal{A}(S{C}_{\gamma }^l)$ to separate $({\mathbb{R}}^2,\mathcal{U})$ into exactly two non-empty components, compared to a typical Jordan curve theorem on $({\mathbb{R}}^2,\mathcal{U})$. Since not every $S{C}_{\gamma }^l$ always separates ( ${\mathbb{Z}}^2,\gamma )$ into two nonempty components, we find a condition for $S{C}_{\gamma }^l,l\ne 4,$ to separate $({\mathbb{Z}}^2,\gamma )$ into exactly two components. The semi-Jordan curve theorem on the $MW$-topological plane plays an important role in applied topology such as digital topology, mathematical morphology as well as computer science.

The purpose of the paper is to study digital topological versions of typical topological groups. In relation to this work, given a digital image $(X,k),X\subset {\mathbb{Z}}^n$, we are strongly required to establish the most suitable adjacency relation in a digital product $X\times X$, say $G_{k^{\ast }}$, that supports both $G_{k^{\ast }}$-connectedness of $X\times X$ and $(G_{k^{\ast }},k)$-continuity of the multiplication $\alpha :(X\times X,G_{k^{\ast }})\to (X,k)$ for formulating a digitally topological $k$-group (or $DT$- $k$-group for brevity). Thus the present paper studies two kinds of adjacency relations in a digital product such as a $C_{k^{\ast }}$- and $G_{k^{\ast }}$-adjacency. In particular, the $G_{k^{\ast }}$-adjacency relation is a new adjacency relation in $X\times X\subset {\mathbb{Z}}^{2n}$ derived from $(X,k)$. Next, the paper initially develops two types of continuities related to the above multiplication $\alpha$, e.g., the $(C_{k^{\ast }},k)$- and $(G_{k^{\ast }},k)$-continuity. Besides, we prove that while the $(C_{k^{\ast }},k)$-continuity implies the $(G_{k^{\ast }},k)$-continuity, the converse does not hold. Taking this approach, we define a $DT$- $k$-group and prove that the pair $(S{C}_k^{n,l},\ast )$ is a $DT$- $k$-group with a certain group operation $\ast$, where $S{C}_k^{n,l}$ is a simple closed $k$-curve with $l$ elements in ${\mathbb{Z}}^n$. Also, the $n$-dimensional digital space $({\mathbb{Z}}^n,2n,+)$ with the usual group operation " $+$" on ${\mathbb{Z}}^n$ is a $DT$- $2n$-group. Finally, the paper corrects some errors related to the earlier works in the literature.