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