The interior of a set is the collection of all points within the set that do not touch the boundary of the set. The closure of a set is the union of the set and its boundary points. The boundary of a set is the collection of points that are both in the set and not in the interior of the set. Cluster points, also known as limit points, are points that are arbitrarily close to the set, but not necessarily in the set.
The interior and closure points are complementary to each other, meaning that the interior points are not in the closure points and vice versa. The boundary points contain elements from both the interior and closure points. Cluster points are a subset of the boundary points, as they are also included in the closure.
These points are important in understanding the structure and properties of a set. They help to define the boundary of a set and determine if the set is open or closed. Cluster points are particularly useful in studying limits and continuity in calculus.
To identify the interior points, we can use the definition that they are points within the set that do not touch the boundary. The closure points can be found by taking the union of the set and its boundary points. Boundary points can be identified by finding points that are both in the set and not in the interior. Cluster points can be determined by finding points that are arbitrarily close to the set.
In an open set, all of the points are interior points and there are no boundary points. The closure of an open set is equal to the set itself. In a closed set, all of the points are either closure points or boundary points. The interior of a closed set may be empty, and the closure is equal to the set itself. Cluster points can exist in both open and closed sets.