A topological space X X is said to be e e -compact with respect to a dense subset D D provided either of the following equivalent conditions is satisfied: (i) every open cover of X X has a finite subcollection which covers D D ; (ii) every ultrafilter on D D converges to a point of X X . If there exists a dense subset with respect to which a space X X is e e -compact, then X X is called e e -compact. 1 ^{1} Two problems recently raised by S. H. Hechler are the following. (a) Is every minimal Hausdorff space e e -compact? (b) If there exists a Hausdorff space which is e e -compact with respect to a space D D , must D D be completely regular? The main purpose of this paper is to provide a negative answer to (a) and to present some results which the author hopes will be of use in the solution to (b). These results can also be used to obtain a construction of ÎČ X \beta X for certain completely regular Hausdorff spaces X X .