An open ball, also known as a deleted neighborhood, is a fundamental concept in topology and metric spaces. It refers to a set of points contained within a given distance from a central point, all lying within the same space.
Formally, in a metric space (X, d), where X denotes the underlying set and d represents the metric, an open ball B(x, r) is defined as the set of all points y in X that are at a distance less than r from a given point x. Mathematically, this can be represented as:
B(x, r) = {y ∈ X : d(x, y) < r}
An open ball is characterized by its center (x) and its radius (r). It is called "open" because it excludes its boundary. In other words, the points lying exactly at a distance r from x are not included in the open ball.
The concept of an open ball is a fundamental tool used in the definition of many concepts in topology and analysis. It is used to define open sets, neighborhoods, limit points, and continuity, among others. Open balls provide a way to describe and explore the local properties of a topological space, enabling the study of convergence, connectedness, and other essential properties.
In summary, an open ball is a set of points in a metric space that are located within a specific distance from a given point, excluding the boundary. It forms a crucial building block in the development and understanding of topological spaces.
