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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Compact spaces and products of finite spaces

Author: Douglas Harris
Journal: Proc. Amer. Math. Soc. 35 (1972), 275-280
MSC: Primary 54D30
MathSciNet review: 0298622
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Abstract: It is shown that the compact spaces are precisely the extension closed subspaces of products of finite spaces, where a subspace is extension closed if every open cover of the subspace extends to an open cover of the entire space. Every closed subspace is extension closed, and for Hausdorff spaces the converse also holds. A single compact space U is constructed, such that every compact space is an extension closed subspace of a product of copies of U; this parallels precisely the property possessed by the unit interval with respect to compact Hausdorff spaces.

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
  • [2] Douglas Harris, Extension closed and cluster closed subspaces, Canad. J. Math. 24 (1972), 1132–1136. MR 0312466
  • [3] -, Universal compact $ {T_1}$ spaces, General Topology and Appl. (to appear).

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Keywords: Extension closed subspace, universal compact space, degree of completeness
Article copyright: © Copyright 1972 American Mathematical Society