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The class of compact* spaces is productive and closed hereditary


Author: W. Govaerts
Journal: Proc. Amer. Math. Soc. 66 (1977), 167-168
MSC: Primary 54D30
DOI: https://doi.org/10.1090/S0002-9939-1977-0461432-6
MathSciNet review: 0461432
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Abstract: W. W. Comfort defined compact$ ^ \ast $ spaces as completely regular Hausdorff spaces X such that each maximal ideal in the ring $ {C^ \ast }(X,{\mathbf{R}})$ of bounded continuous real-valued functions on X is fixed. He showed that, independently of the axiom of choice, the class of compact$ ^ \ast $ spaces is productive and closed hereditary. We give a short new proof of this.


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

  • [1] W. W. Comfort, A theorem of Stone-Čech type, and a theorem of Tychonoff type, without the axiom of choice; and their realcompact analogues, Fund. Math. 63 (1968), 97-110. MR 0236880 (38:5174)
  • [2] W. Govaerts, Representation and determination problems: a case study, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24 (1976), 57-59. MR 0402675 (53:6491)
  • [3] -, A Separation axiom for the study of function space structures, ibid. 24 (1976), 65-69. MR 0402676 (53:6492)
  • [4] S. Salbany, On compact$ ^\ast$ spaces and compactifications, Proc. Amer. Math. Soc. 45 (1974), 274-280. MR 0355970 (50:8443)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0461432-6
Keywords: Compact$ ^ \ast $ spaces, E-compactness, ring of continuous functions, representation of homomorphisms
Article copyright: © Copyright 1977 American Mathematical Society

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