Semirings of functions determine finite 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 topological spaces. Let be the semiring of continuous functions from finite *X* into *Z*, the nonnegative integers with open sets of the form . Products and sums in are defined pointwise. Denote the set of nonzero semiring homomorphisms of into *Z* by and give it the compact-open topology where is considered discrete. Then (1) *X* and are homeomorphic. (2) is semiring isomorphic to iff *X* is homeomorphic to *Y*. (3) The topology of *X* can be completely recovered from the inclusion relations among the ideals of which are kernels of the elements in .

**[1]**V. G. Boltjanskiĭ,*Separability axioms and metric*, Dokl. Akad. Nauk SSSR**197**(1971), 1239–1242 (Russian). MR**0281145****[2]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[3]**Michael C. McCord,*Singular homology groups and homotopy groups of finite topological spaces*, Duke Math. J.**33**(1966), 465–474. MR**0196744****[4]**R. E. Stong,*Finite topological spaces*, Trans. Amer. Math. Soc.**123**(1966), 325–340. MR**0195042**, https://doi.org/10.1090/S0002-9947-1966-0195042-2

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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