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In topology, the boundary of a set X is the intersection of the closure of X and the closure of the complement of X.

Every set has a boundary, but only closed sets contain their boundaries. In contrast, an open set never contains its boundary. Example: both [0,1] and (0,1) have the boundary {0,1}. The first set is closed and contains its boundary; the second set is open and does not contain its boundary. Depending on the context, boundaries can be referred to by more common names: perimeter, surface, frontier, border, and edge.