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Proceedings of the American Mathematical Society

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



On $ \mu $-spaces and $ k\sb{R}$-spaces

Author: J. L. Blasco
Journal: Proc. Amer. Math. Soc. 67 (1977), 179-186
MSC: Primary 54D15
MathSciNet review: 0464152
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Abstract: In this paper it is proved that when X is a $ {k_R}$-space then $ \mu X$ (the smallest subspace of $ \beta X$ containing X with the property that each of its bounded closed subsets is compact) also is a $ {k_R}$-space; an example is given of a $ {k_R}$-space X such that its Hewitt realcompactification, $ \upsilon X$, is not a $ {k_R}$-space. We show with an example that there is a non-$ {k_R}$-space X such that $ \upsilon X$ and $ \mu X$ are $ {k_R}$-spaces. Also we answer negatively a question posed by Buchwalter: Is $ \mu X$ the union of the closures in $ \upsilon X$ of the bounded subsets of X? Finally, without using the continuum hypothesis, we give an example of a locally compact space X of cardinality $ {\aleph _1}$ such that $ \upsilon X$ is not a k-space.

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Keywords: $ \mu $-space, $ {k_R}$-space, k-space, bounded subset, $ {k_R}X,kX,\mu X$, P-space, bornological space, barrelled space, ultrabornological space, complete space, compact-open topology, measurable cardinal
Article copyright: © Copyright 1977 American Mathematical Society

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