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A boundary encloses a region of space, a territory, and/or an area. Something enclosed by a boundary, is "bounded" (by the boundary).

See also border

In topology and related fields, the boundary of a subset is the set of all points of that space in its closure that are not interior points. More precisely, the boundary of a subset S is the difference between its closure and its interior. The boundary is always a closed set.

On the other hand, a manifold with boundary means something like a manifold, but modelled on a half-Euclidean space, on one side of a hyperplane. If the boundary points are removed an open subset that is a genuine manifold remains. The boundaries of manifolds are important, for example, in Stokes' theorem, and cobordism theory.