A space in which every infinite set contains an infinite subset with only a
finite number of accumulation  points is said to have the   finite derived
set property. We  study this  property in  the  class of  spaces in  which
compact sets are closed --  the    KC-spaces  -- and apply our results to
show that among hereditarily Lindel\"of  spaces,  minimal   KC-spaces are
compact. This result generalizes a theorem of \cite{AW} and gives a partial
answer to a question of R. Larson.