Inverse compactness

AbstractNew classes of spaces between compact and countably compact are considered. A space X is inversely compact provided every independent family of closed subsets of X has nonempty intersection. In other words, X is inversely compact iff for every open cover u of X, one can select a finite cover of X consisting of elements of u and their supplements. Similar modifications of other compact-type properties are considered. A T1, space is inversely countably compact iff it is countably compact. An example of a T1 inversely compact, noncompact space is given