In general we simplify the definition to be: A compact set is a set which is closed (that is it contains its boundary points) and is bounded.
Example [2,8] is a compact set. The unit disk including the boundary is a compact set. (3,5] is not a compact set. Note that all of these examples are of sets that are uncountably infinite.
Technical def: A set is said to be compact if for every open ["Cover"] there exists a finite SubCover which also covers the set.