Boundary point

In geometry and, more generally, in topology, a boundary point of a set (figure, body) is a point of the space such that in every neighbourhood there are points which belong to the set and points which do not belong to the set.

A boundary point may or may not belong to the set. A point of the set which is not a boundary point is called interior point. A point not in the set which is not a boundary point is called exterior point.

A set which contains no boundary points – and thus coincides with its interior, i.e., the set of its interior points – is called open.

A set which contains all its boundary points – and thus is the complement of its exterior – is called closed.

The set of all boundary points of a set S is called the boundary of the set. For a set in the plane, its length – if it is defined – is called the perimeter of the set.

The boundary of 3-dimensional body is also called its surface, and its area – if it is defined – is called the surface area.