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)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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$.


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

  • [1] Ryszard Engelking, Teoria wymiaru, Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51]. MR 0482696
    Ryszard Engelking, Dimension theory, North-Holland Publishing Co., Amsterdam-Oxford-New York; PWN—Polish Scientific Publishers, Warsaw, 1978. Translated from the Polish and revised by the author; North-Holland Mathematical Library, 19. MR 0482697
  • [2] V. V. Filippov, Bicompacta with distinct inductive dimensions, Dokl. Akad. Nauk SSSR 192 (1970), 289–292 (Russian). MR 0266173
  • [3] V. V. Filippov, The inductive dimension of a product of bicompacta, Dokl. Akad. Nauk SSSR 202 (1972), 1016–1019 (Russian). MR 0292043
  • [4] R. Galián, Teoría de la dimensión, Serie Univ. de la Fund. Juan March, no. 107, Madrid, 1979.
  • [5] Leonard Gillman and Meyer Jerison, Rings of continuous functions, Springer-Verlag, New York-Heidelberg, 1976. Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43. MR 0407579
  • [6] J. Isbell, Graduation and dimension in locales, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 195–210. MR 787829
  • [7] Miroslav Katětov, A theorem on the Lebesgue dimension, Časopis Pěst. Mat. Fys. 75 (1950), 79–87 (English, with Czech summary). MR 0036502
  • [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] Ma. Teresa Sancho de Salas, Methods of commutative algebra for topology, Publicaciones del Departamento de Matemáticas, Universidad de Extremadura [Publications of the Mathematics Department of the University of Extremadura], vol. 17, Universidad de Extremadura, Facultad de Ciencias, Departamento de Matemáticas, Badajoz; Universidad de Salamanca, Departamento de Matemáticas, Salamanca, 1987. MR 932467

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C40, 46J10, 54F45

Retrieve articles in all journals with MSC: 54C40, 46J10, 54F45


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