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Keimel and Lawson proposed a set of conditions for proving a category oftopological spaces to be reflective in the category of all T0 spaces. Theseconditions were recently used to prove the reflectivity of the category of allwell-filtered spaces. In this paper, we prove that, in certain sense, theseconditions are not just sufficient but also necessary for a category of T0spaces to be reflective. Using this general result, we easily deduce thatseveral categories proposed in domain theory are not reflective, thus answereda few open problems.

The reflectivity of some categories of T0 spaces in domain theory