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Proceedings of the American Mathematical Society

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A note on $ \alpha $-compact spaces


Author: Teklehaimanot Retta
Journal: Proc. Amer. Math. Soc. 89 (1983), 314-316
MSC: Primary 54B10; Secondary 03E55, 54D60
MathSciNet review: 712643
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Abstract: For an infinite cardinal $ \alpha $, $ m(\alpha )$ denotes the least measurable cardinal, if one exists, not less than $ \alpha $. We give easy proofs of generalizations of some results on realcompact spaces. Among these we prove the following generalization of a theorem of A. Kato.

Let $ \{ {X_i}:i \in I\} $ be a collection of spaces each having at least two elements. Then the $ k$-box product $ {(\prod {X_i})_k}$ is $ \alpha $-compact if and only if either $ {X_i}$ is $ \alpha $-compact for each $ i \in I$ and $ k \leqslant m(\alpha )$ or $ \left\vert I \right\vert < m(\alpha )$.


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

  • [1] W. Wistar Comfort and Teklehaimanot Retta, Generalized perfect maps and a theorem of I. Juhász, Rings of continuous functions (Cincinnati, Ohio, 1982) Lecture Notes in Pure and Appl. Math., vol. 95, Dekker, New York, 1985, pp. 79–102. MR 789263
  • [2] Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
  • [3] Akio Kato, Realcompactness of box products, Mem. Defense Acad. 19 (1979), no. 1, 1–4. MR 532792
  • [4] T. Retta, Doctoral Dissertation, Wesleyan University, 1977.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0712643-6
Keywords: $ \alpha $-compact space, measurable cardinal, box topology, $ \kappa $-box topology, $ \alpha $-base
Article copyright: © Copyright 1983 American Mathematical Society