The Avoidance Spectrum of Alexandroff Spaces

Abstract

In this paper we prove that every T0 Alexandroff topological space (𝑋, 𝜏) is homeomorphic to the avoidance of a subspace of (Spec(Λ), 𝜏𝑍), where Spec(Λ) denotes the prime spectrum of a semi-ring Λ induced by 𝜏, and 𝜏𝑍 is the Zariski topology. We also prove that (Spec(Λ), 𝜏𝑍) is an Alexandroff space if and only if Λ satisfies the Gilmer property.

Keywords

• alexandroff space; avoidance spectrum; zariski topology.