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Proceedings of the American Mathematical Society

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

 

 

A remark on irreducible spaces


Author: J. C. Smith
Journal: Proc. Amer. Math. Soc. 57 (1976), 133-139
MSC: Primary 54D20
MathSciNet review: 0405353
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Abstract: A topological space $ X$ is called irreducible if every open cover of $ X$ has an open refinement which covers $ X$ minimally. In this paper we show that weak $ \bar \theta $-refinable spaces are irreducible. A modification of the proof of this result then yields that $ {\aleph _1}$-compact, weak $ \overline {\delta \theta } $-refinable spaces are Lindelöf. It then follows that perfect, $ {\aleph _1}$-compact weak $ \delta \theta $-refinable spaces are irreducible. A number of known results follow as corollaries.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0405353-2
Keywords: Metacompact, $ \theta $-refinable, weak $ \bar \theta $-refinable, $ \delta \theta $-refinable, weak $ \overline {\delta \theta } $-refinable, $ {\aleph _1}$-compact, compact, countably compact, Lindelöf, minimal cover, irreducible, maximal distinguished set, perfect
Article copyright: © Copyright 1976 American Mathematical Society