The Avoidance Spectrum of Alexandroff Spaces
Published
May 4, 2024
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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.
References
[1] Gilmer R. An intersection condition for prime ideals, chapter 5 in Factorization in integral domains, edited by D. Anderson, eBook, Routledge, New York, 2017
[2] Golan J. Semirings and their applications. Kluwer Academic Publishers, Dordrecht, 1999
[3] Hungerford T. Algebra. Springer-Verlag, New York, 2000
[4] Lang S. Algebra. Springer-Verlag, New York, 2002
[5] Munkres J. Topology. Prentice Hall, Upper Saddle River, 2000
[6] Uzcátegui C. and Vielma, J. Alexandroff topologies viewed as closed sets in the Cantor cube, Divulgaciones Matemáticas, 13 (2): 33–45, 2005
[2] Golan J. Semirings and their applications. Kluwer Academic Publishers, Dordrecht, 1999
[3] Hungerford T. Algebra. Springer-Verlag, New York, 2000
[4] Lang S. Algebra. Springer-Verlag, New York, 2002
[5] Munkres J. Topology. Prentice Hall, Upper Saddle River, 2000
[6] Uzcátegui C. and Vielma, J. Alexandroff topologies viewed as closed sets in the Cantor cube, Divulgaciones Matemáticas, 13 (2): 33–45, 2005
How to Cite
Mejias, L., & Vielma, J. E. (2024). The Avoidance Spectrum of Alexandroff Spaces. Universitas Scientiarum, 29(2), 97–106. https://doi.org/10.11144/Javeriana.SC292.taso
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Mathematics and Statistics
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