  Keimel and Lawson proposed a set of conditions for proving a category of
topological spaces to be reflective in the category of all T0 spaces. These
conditions were recently used to prove the reflectivity of the category of all
well-filtered spaces. In this paper, we prove that, in certain sense, these
conditions are not just sufficient but also necessary for a category of T0
spaces to be reflective. Using this general result, we easily deduce that
several categories proposed in domain theory are not reflective, thus answered
a few open problems.
