That is, given any collection of open sets whose union contains A, only a finite number of the open sets are actually needed to cover A; the rest can be thrown out.
For sets contained in a Euclidean space, such as R^3, a set is compact if, and only if, it is closed (in the sense of point-set topology) and bounded (i.e., is contained in some sphere).
For example, a torus or a cube is compact, but a plane in space is not, since it is not bounded.
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The Geometry Center