Construction and formal verification of a special topology on {\mathbb{Z}}^{n}: characterization of connected subsets

Abstract

We examine whether the integers admit a topology whose connected subsets are precisely the sets of consecutive integers. The main result gives a construction for every integer lattice {\mathbb{Z}}^{n}, n>0; the answer for Z is then obtained as a corollary. We prove that the neighborhoods determined by the parity of the sum of the coordinates form a basis and that topological connectedness agrees exactly with accessibility by finite paths of unit lattice steps. We also present a deductive audit and the correspondence between this argument and a verification in Lean 4 and Mathlib, including the empty set and the independent formal characterization of integer intervals. The construction and mathematical result originate in Problem 5712, proposed by Dan Marcus and solved in 1970 by the Cleveland State University group; the present article contributes a detailed exposition and a report on formal verification.