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In two dimensions, closed disks are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary.
However, an open disk is not compact, because a sequence of points can tend to the boundary without getting arbitrarily close to any point in the interior.
Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can tend to the missing point without tending to any point within the space.
Lines and planes are not compact, since one can take a set of equally spaced points in any given direction without approaching any point.

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