Let be a topological space with topology , and be a subset of . A point is said to be the limit point or accumulation point or cluster point of if each open set containing contains at least one point of different from .
In other words, a point of a topological space is said to be the limit point of a subset of if for every open set containing we have
It is clear from the above definition that the limit point of a set may or may not be the point of .
Let with topology and, then is the only limit point of , because the open sets containing , namely and , also contain a point of .
On the other hand, and are not limit points of , because the open set containing these points does not contain any points of . Point is also not a limit point of , because the open set containing does not contain any other point of different from . Thus, the set has no limit points.
As another example, let with topology . Let then is not a limit point of , because the open set containing does not contain any other point of different from . is a limit point of , because the open sets and containing also contain a point of different from . Similarly, and are also limit points of . This illustration suggests that a set can have more than one limit point.
Let be a topological space, and let be a subset of . The set of all limit points of is said to be the derived set and is denoted by or . In the above example, .
It may be noted that under usual topology, consider the subsets , , , of real, then all the points of these intervals are limits points.