The term you're looking for is "simply-connected set." Simply-connected sets and compact sets are both sometimes informally described as "sets with no holes,"

The term you're looking for is "simply-connected set." Simply-connected sets and compact sets are both sometimes informally described as "sets with no holes," but the definition of "holes" is different in both cases. A compact set contains all of its limiting points (i.e. limits of convergent sequences of points within the set). Therefore it may not have any point-shaped holes. But it may still have a hole shaped like, e.g. an open disc. To rule out such large holes, you want a simply-connected set, which is a connected set in which any closed path can be continuously shrunk to a point.