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Semirings of functions determine finite $ T\sb{o}$ topologies


Author: Melvin C. Thornton
Journal: Proc. Amer. Math. Soc. 34 (1972), 307-310
MSC: Primary 54A10; Secondary 54C40
DOI: https://doi.org/10.1090/S0002-9939-1972-0292019-1
MathSciNet review: 0292019
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Abstract: An analogue of the Stone-Gelfand-Kolmogoroff theorem for compact Hausdorff spaces is proven for finite $ {T_0}$ topological spaces. Let $ C(X)$ be the semiring of continuous functions from finite $ {T_0}$ X into Z, the nonnegative integers with open sets of the form $ \{ 0,1,2, \cdots ,m\} $. Products and sums in $ C(X)$ are defined pointwise. Denote the set of nonzero semiring homomorphisms of $ C(X)$ into Z by $ H(X)$ and give it the compact-open topology where $ C(X)$ is considered discrete. Then (1) X and $ H(X)$ are homeomorphic. (2) $ C(X)$ is semiring isomorphic to $ C(Y)$ iff X is homeomorphic to Y. (3) The topology of X can be completely recovered from the inclusion relations among the ideals of $ C(X)$ which are kernels of the elements in $ H(X)$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0292019-1
Keywords: Finite space, semirings, rings of continuous functions
Article copyright: © Copyright 1972 American Mathematical Society

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