  Ern\'e weakened the concept of sobriety in order to extend the theory of
sober spaces and locally hypercompact spaces to situations where directed joins
were missing, and introduced and discussed three kinds of non-sober spaces: cut
spaces, weakly sober spaces, and quasisober spaces. Three other kinds of
non-sober spaces, namely $\mathsf{DC}$ space, $\mathsf{RD}$ space and
$\mathsf{WD}$ space, were introduced and investigated by Xu, Shen, Xi and Zhao.
All these six kinds of spaces are strictly weaker than sober spaces. In this
paper, it is shown that none of the category of all $\mathsf{DC}$ spaces, that
of all $\mathsf{RD}$ spaces, that of all $\mathsf{WD}$ spaces, that of all
quasisober spaces, that of all weakly spaces and that of all cut spaces is
reflective in the category of all $T_0$ spaces with continuous mappings.
