Boundary point: Difference between revisions
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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 (topology)|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''. | |||
Revision as of 18:02, 30 September 2009
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.