  In this paper, we obtain some sufficient conditions for the D-completion of a
T0 space to be the well-filterification of this space, the well-filterification
of a T0 space to be the sobrification of this space and the D-completion of a
T0 space to be the sobrification, respectively. Moreover, we give an example to
show that a tapered closed set may be neither the closure of a directed set nor
the closed KF-set, respectively. Because the tapered closed set is a closed
WD-set, the example also gives a negative answer to a problem proposed by Xu.
Meantime, a new direct characterization of the D-completion of a T0 space is
given by using the notion of pre-c-compact elements.
