Semirings of functions determine finite topologies
Melvin C. Thornton
Proc. Amer. Math. Soc. 34 (1972), 307-310
Primary 54A10; Secondary 54C40
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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 .
G. Boltjanskiĭ, Separability axioms and metric, Dokl.
Akad. Nauk SSSR 197 (1971), 1239–1242 (Russian). MR 0281145
Dugundji, Topology, Allyn and Bacon Inc., Boston, Mass., 1966.
0193606 (33 #1824)
C. McCord, Singular homology groups and homotopy groups of finite
topological spaces, Duke Math. J. 33 (1966),
465–474. MR 0196744
E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), 325–340. MR 0195042
(33 #3247), http://dx.doi.org/10.1090/S0002-9947-1966-0195042-2
- V. G. Boltjanskii, Separability axioms and a metric, Dokl. Akad. Nauk SSSR 197 (1971), 1239-1242 = Soviet Math. Dokl. 12 (1971), 639-643. MR 0281145 (43:6864)
- J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
- M. C. McCord, Singular homology groups and homotopy groups of finite topological spaces, Duke Math. J. 33 (1966), 465-474. MR 33 #4930. MR 0196744 (33:4930)
- R. E. Stong, Finite topological spaces, Trans. Amer. Math. Soc. 123 (1966), 325-340. MR 33 #3247. MR 0195042 (33:3247)
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