Structure spaces for rings of continuous functions with applications to realcompactifications

Lothar Redlin; Saleem Watson

Fundamenta Mathematicae (1997)

  • Volume: 152, Issue: 2, page 151-163
  • ISSN: 0016-2736

Abstract

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Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).

How to cite

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Redlin, Lothar, and Watson, Saleem. "Structure spaces for rings of continuous functions with applications to realcompactifications." Fundamenta Mathematicae 152.2 (1997): 151-163. <http://eudml.org/doc/212203>.

@article{Redlin1997,
abstract = {Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).},
author = {Redlin, Lothar, Watson, Saleem},
journal = {Fundamenta Mathematicae},
keywords = {ring of continuous functions; maximal ideal; ultrafilter; realcompactification; local bounded inversion; Stone-Čech compactification; structure space},
language = {eng},
number = {2},
pages = {151-163},
title = {Structure spaces for rings of continuous functions with applications to realcompactifications},
url = {http://eudml.org/doc/212203},
volume = {152},
year = {1997},
}

TY - JOUR
AU - Redlin, Lothar
AU - Watson, Saleem
TI - Structure spaces for rings of continuous functions with applications to realcompactifications
JO - Fundamenta Mathematicae
PY - 1997
VL - 152
IS - 2
SP - 151
EP - 163
AB - Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).
LA - eng
KW - ring of continuous functions; maximal ideal; ultrafilter; realcompactification; local bounded inversion; Stone-Čech compactification; structure space
UR - http://eudml.org/doc/212203
ER -

References

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  1. [1] W. Adamski, Two ultrafilter properties for vector lattices of real-valued functions, Publ. Math. Debrecen 45 (1994), 225-267. Zbl0833.46015
  2. [2] R. M. Brooks, A ring of analytic functions, Studia Math. 24 (1964), 191-210. Zbl0199.46201
  3. [3] H. L. Byun, L. Redlin and S. Watson, Local invertibility in subrings of C*(X), Bull. Austral. Math. Soc. 46 (1992), 449-458. 
  4. [4] H. L. Byun and S. Watson, Prime and maximal ideals in subrings of C(X), Topology Appl. 40 (1991), 45-62. Zbl0732.54016
  5. [5] L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, New York, 1978. Zbl0093.30001
  6. [6] M. Henriksen, J. R. Isbell and D. G. Johnson, Residue class fields of lattice-ordered algebras, Fund. Math. 50 (1961), 107-117. Zbl0101.33401
  7. [7] M. Henriksen and D. G. Johnson, On the structure of a class of archimedean lattice-ordered algebras, Fund. Math. 50 (1961), 73-94. Zbl0099.10101
  8. [8] D. Plank, On a class of subalgebras of C(X) with applications to βX, Fund. Math. 64 (1969), 41-54. 
  9. [9] L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math. Soc. 100 (1987), 763-766. Zbl0622.54011
  10. [10] S. Willard, General Topology, Addison-Wesley, Reading, Mass., 1970. 

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