Understanding Neighborhoods, Interior, Exterior, and Boundary Points in Geometry

Neighborhood

A set of points is called a neighborhood of a point. For example, in the plane or in space one can define such neighborhoods. If the radius is not emphasized, the notation can be simplified. In practice square neighborhoods are often used because they can contain circular neighborhoods and vice versa.

Region

(1) Interior, exterior, and boundary points

If there exists a neighborhood of a point that is entirely contained in a set, the point is an interior point of the set.

If there exists a neighborhood of a point that is entirely contained in the complement of the set, the point is an exterior point of the set.

If every neighborhood of a point contains both interior points and exterior points of the set, the point is a boundary point. Interior points belong to the set, exterior points do not, and boundary points may or may not belong.

(2) Accumulation point

If for any given punctured neighborhood of a point there is always a point of the set inside it, the point is an accumulation point of the set. An accumulation point may belong to the set or not. The collection of all accumulation points is called the derived set of the original set.

(3) Open and closed regions

If all points of a set are interior points, the set is an open set.

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

If a set contains all its limit points, it is a closed set.

If any two points in a set can be connected by a polyline that lies entirely within the set, the set is connected.

A connected open set is called an open region (simply “region”).

An open region together with its boundary is called a closed region.

n‑Dimensional Space

An ordered n‑tuple of real numbers forms the n‑dimensional Euclidean space, denoted \(\mathbb{R}^n\). Each element is a point, and its components are the coordinates. When all coordinates are zero, the point is the zero vector.

The distance between two points in \(\mathbb{R}^n\) is defined by the Euclidean norm, and the distance from a point to the zero vector is its norm.

Definition 1 . For a non‑empty set of points \(A\), a mapping \(f\) defined on \(A\) is called a function on \(A\). The set \(A\) is the domain of the function, and the set of its values is the codomain. Special cases include binary functions (two variables) and ternary functions (three variables).

For example, a binary function whose domain is a circular region may have its graph as the upper hemisphere of a sphere centered at the origin. Generally, the graph of a binary function is a surface in space. A ternary function with domain the unit closed ball has a graph that is a hypersurface in \(\mathbb{R}^3\).