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Dimension of dense subalgebras of $ C(X)$


Authors: Juan B. Sancho de Salas and Ma. Teresa Sancho de Salas
Journal: Proc. Amer. Math. Soc. 105 (1989), 491-499
MSC: Primary 54C40; Secondary 46J10, 54F45
DOI: https://doi.org/10.1090/S0002-9939-1989-0929426-X
MathSciNet review: 929426
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Abstract: The real spectrum of any $ {\mathbf{R}}$-algebra $ A$ is the set of all maximal ideals of $ A$ with residue field $ {\mathbf{R}}$, endowed with the initial topology for the functions induced by the elements of $ A$. We prove that a compact metric space $ X$ has dimension $ \leq n$ if and only if $ X$ is the real spectrum of an algebra of Krull dimension $ \leq n$; so that the dimension of $ X$ is the minimum of the Krull dimensions of all dense subalgebra of $ C(X)$. Moreover, we prove that a compact Hausdorff space $ X$ has covering dimension $ \leq n$ if and only if every countably generated subalgebra of $ C(X)$ is contained in the closure of a subalgebra of Krull dimension $ \leq n$.


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  • [1] R. Engelking, Dimension theory, North-Holland Math. Library, vol. 19, North-Holland, Amsterdam, 1978. MR 0482697 (58:2753b)
  • [2] V. V. Filippov, On bicompacta with noncoinciding inductive dimensions, Dokl. Akad. Nauk SSSR 192 (1970), 289-292; English transi., Soviet Math. Dokl. 11 (1970), 635-638. MR 0266173 (42:1081)
  • [3] -, On the inductive dimension of the product of bicompacta, Dokl. Akad. Nauk SSSR 202 (1972), 1016-1019; English transl., Soviet Math. Dokl. 13 (1972), 250-254. MR 0292043 (45:1131)
  • [4] R. Galián, Teoría de la dimensión, Serie Univ. de la Fund. Juan March, no. 107, Madrid, 1979.
  • [5] L. Gillman and M. Jerison, Rings of continuous functions, Graduate Texts in Math., no. 43, Springer-Verlag, Heidelberg, 1976. MR 0407579 (53:11352)
  • [6] J. Isbell, Graduation and dimension in locales, London Math. Soc. Lecture Notes Ser., no. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 195-210. MR 787829 (86h:54042)
  • [7] M. Katětov, A theorem on the Lebesgue dimension, Časopis Pěst. Mat. Fys. 75 (1950), 79-87. MR 0036502 (12:119c)
  • [8] J. A. Navarro, Espacios topológicos finitos y homotopía, VIII. Jornadas Luso-Espanholas de Mat., Univ. de Coimbra, 1981, pp. 331-337.
  • [9] J. B. Sancho and M. T. Sancho, Some results on dimension theory, Summer Meeting on Category Theory, Louvain-la-Neuve, 1987.
  • [10] M. T. Sancho, Methods of commutative algebra for topology, Publ. Mat., no. 17, Univ. de Extremadura, Badajoz, 1987. MR 932467 (89f:54003)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0929426-X
Keywords: Dense subalgebras of $ C(X)$, Krull dimension, real spectrum, inverse limits of polyhedra, covering dimension
Article copyright: © Copyright 1989 American Mathematical Society

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