``cofinite sieves''

Have any topos-theorists or others come across the following notion? Let C be a small category, and c an object of C. If X is an arbitrary set of arrows with codomain c, then R_X = { f:b-->c | there is no g:a-->b with fg in X} clearly gives a sieve on c. Of course if X itself were a sieve then R_X would be its complement, but I'm not assuming X is a sieve. Say that a sieve R is _cofinite_ if it is of the form R_X for a finite set X. Cofinite sieves are closed under finite intersection and universal quantification along an arbitrary arrow. Best wishes, Steve Lack.