Abstract
The latticeA(X) of all possible subalgebras of the ring of all continuous ℝ-valued functions defined on an ℝ-separated spaceX is considered. A topological space is said to be a Hewitt space if it is homeomorphic to a closed subspace of a Tychonoff power of the real line ℝ. The main achievement of the paper is the proof of the fact that any Hewitt spaceX is determined by the latticeA(X). An original technique of minimal and maximal subalgebras is applied. It is shown that the latticeA(X) is regular if and only ifX contains at most two points.
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References
L. Gillman and M. Jerison,Rings of Continuous Functions, Springer-Verlag, New York (1976).
E. M. Vechtomov, “Questions on the determination of topological spaces by algebraic systems of continuous functions,” in:Algebra. Topologiya. Geometriya. Itogi Nauki i Tekhniki [in Russian], Vol. 28, VINITI, Moscow (1990), pp. 3–46.
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Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 687–693, November, 1997.
Translated by A. I. Shtern
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Vechtomov, E.M. Lattice of subalgebras of the ring of continuous functions and Hewitt spaces. Math Notes 62, 575–580 (1997). https://doi.org/10.1007/BF02361295
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DOI: https://doi.org/10.1007/BF02361295