The group of autohomeomorphisms of some digital topologies on the integers

For a topological space X, the group of autohomeomorphisms is denoted by H(X). It is a wellestablished 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, H(X), acts transitively on X, via the action H(X) ? X ?? X (1, x) 7?? 1(x). The degree of homogeneity of X, denoted as dH(X), is defined as the cardinality of the quotient set X/H(X) relative to the aforementioned action. Regarding the Khalimsky topology defined on the set of integers, this topology, denoted by K, is the topology generated by the family {{x ? 1, x, x + 1} : x is an even integer}, as a subbase. The space (Z,K), known as the Khalimsky line or digital line, will be denoted by KL (or KL1). The digital line is notably influential in digital image processing and computer graphics. For recent advancements in digital topologies on Zn, see [3], [10], [11], and [12]. The aim of this paper is the construction of a sequence of Alexandroff topologies, {Kp : p ? N}, on the set of integers Z. This provides new digital topologies with the following properties: - H(Z,Kp) is isomorphic to H(Z,K). - For each positive integer p, (Z,Kp) is topologically embedded in (Z,Kp+1). The degree of homogeneity, Krull dimension, inductive dimension and the height of (Z,Kp) are also computed.