Extensions of Monotone Sets *

Far-reaching generalizations of the extension theorem for monotone sets due to Debrunner and Flor are obtained. In fact, Browder's extension theorem involving Kakutani multimaps is extended to the one involving a large class of “better” admissible maps. Moreover, a sharpened version of the extension theorem for noncompact case due to Lassonde is also obtained. Finally, we mention another generalization of the Debrunner-Flor theorem due to the author