A closed set is a mathematical concept that refers to a set of numbers that contains all its limit points. This means that every possible limit point of the set is also an element of the set itself.
Unlike a closed set, an open set does not contain all its limit points. This means that there may be limit points that are not included in the set. Additionally, the boundary points of an open set are not considered to be part of the set, while they are included in a closed set.
The boundary of a closed set is the set of points that are both in the set and in its complement. In other words, the boundary is the edge or boundary between the set and its complement.
In topology, a closed set is defined as a set that contains all of its limit points. This definition can also be extended to topological spaces, where a set is considered closed if its complement is open.
Some examples of closed sets include the set of all integers, the set of all real numbers, and the set of all points within a closed interval on the number line. In topology, the set of all points within a closed circle or sphere is also a closed set.