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 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, H(X), acts transitively on X, via the action [Formula In Abstract]. The degree of homogeneity of X, denoted as d_H (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 [Formula In Abstract]_, 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, {K_p: p ∈ N}, on the set of integers Z. This provides new digital topologies with the following properties:-H(Z, K_p) is isomorphic to H(Z, K).-For each positive integer p, (Z, K_p) is topologically embedded in (Z, K_p+1). The degree of homogeneity, Krull dimension, inductive dimension and the height of (Z, K_p) are also computed.