# Boundary

''Ths article is about the geometric concept. For boundaries in the sport of cricket, see Boundary (cricket).

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.

An alternate meaning for **boundary** can be a psychological, physical or mental place which we keep between ourselves and others. Sometimes this can be referred to as personal space. Addicts use **boundaries** to keep themselves sober. Some times these may be called bottom lines.

See also bottom line behaviour.

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.

See also compactification.