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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Semirings of functions determine finite $T_{o}$ topologies
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by Melvin C. Thornton PDF
Proc. Amer. Math. Soc. 34 (1972), 307-310 Request permission

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
  • © Copyright 1972 American Mathematical Society
  • 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