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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.
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