On 𝜇-spaces and 𝑘_{𝑅}-spaces

Blasco, J. L.

doi:10.1090/S0002-9939-1977-0464152-7

On $\mu$-spaces and $k_{R}$-spaces
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Proc. Amer. Math. Soc. 67 (1977), 179-186 Request permission

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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