Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



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
MathSciNet review: 0292019
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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)$.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54A10, 54C40

Retrieve articles in all journals with MSC: 54A10, 54C40

Additional Information

Keywords: Finite space, semirings, rings of continuous functions
Article copyright: © Copyright 1972 American Mathematical Society