Let X is a topological space with topology, and A is a subset of X. A point  is said to be the limit point or accumulation point or cluster point of A if each open set containing contains at least one point of A different from.
            In other words, a point of a topological space X is said to be the limit point of a subset A of X if every open set containing, we have 
                                                               
            It is clear from the above definition that the limit point of a set A may or may not be the point of A.
            Let  with topology  and, then is the only limit point of A, because the open sets containing b namely and X also contains a point a of A.

            On the other hand, a and b are not limit point of, because the open set containing these points do not contain any point of C. The point c is also not a limit  point of C, because the open set containing c does not contain any other point of C different from c. Thus, the set has no limit points.
            As another example, let  with topology . Let  then a is not a limit point of A, because the open set containing a does not contain any other point of A different from a. b is a limit point of A, because the open sets and X containing b also contain a point of A different from b. Similarly, d and e are also limit points of A. This illustration suggests that a set can have more than one limit point.

Derived Set:
            Let be a topological space, and let A be a subset of X. The set of all limit points of A is said to be the derived set and is denoted by or . In the above example, .

Remark:
            It may be noted that under usual topology, consider the subsets, , ,  of real, then all the points of these intervals are limits points.