  M. Escard\'o et al. asked whether the core compactly generated topology of a
sober space is again sober and the sobrification of a core compactly generated
space again core compactly generated. In this note, we answer the problem by
displaying a counterexample, which reveals that the core compactly generated
spaces are not closed under sobrifications. Meantime, we obtain that the core
compactly generated spaces are closed under {\omega}-well-filterifications and
D-completions. Furthermore, we find that the core compactly generated topology
of the Smyth power space of a well-filtered space coincides with the Scott
topology. Finally, we provide a characterization for the core-compactness of
core compactly generated spaces.
