In this paper it is proved that when X is a
k
R
{k_R}
-space then
μ
X
\mu X
(the smallest subspace of
β
X
\beta X
containing X with the property that each of its bounded closed subsets is compact) also is a
k
R
{k_R}
-space; an example is given of a
k
R
{k_R}
-space X such that its Hewitt realcompactification,
υ
X
\upsilon X
, is not a
k
R
{k_R}
-space. We show with an example that there is a non-
k
R
{k_R}
-space X such that
υ
X
\upsilon X
and
μ
X
\mu X
are
k
R
{k_R}
-spaces. Also we answer negatively a question posed by Buchwalter: Is
μ
X
\mu X
the union of the closures in
υ
X
\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
ℵ
1
{\aleph _1}
such that
υ
X
\upsilon X
is not a k-space.