Boundary point

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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.

Boundary

The set of all boundary points of a set S is called the boundary of the set.

In elementary geometry, for figures in the plane (like polygons, convex sets, ...) and bodies in the space (like polyhedra, balls, etc.) the boundary corresponds to the intuitive idea of a boundary: In the plane it is a closed curve, and in space it is a closed surface (like the hide of a balloon).

But even in the plane the situation is more complicated than one might expect. Intuitively, it is "evident" that a closed curve which does not intersect itself is the boundary of an interior bounded set which it separates from the (unbounded) exterior. While this statement is indeed true under quite general assumptions (Jordan's curve theorem), its proof is far from trivial even in the "simple" case that the closed curve is a polygon.

For general sets, and in topology, the extreme cases — every point is a boundary point, or there is no boundary point at all — are both possible. In the first case both the set and its complement are dense in the space. In the second case (empty boundary) the set is both open and closed and called clopen (an artificial word obtained by combining closed and open).

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.