Limit Point of a Set

Let X is a topological space with topology \tau , and A is a subset of X. A point x \in X is said to be the limit point or accumulation point or cluster point of A if each open set containingx contains at least one point of A different from x.

In other words, a point x of a topological space X is said to be the limit point of a subset A of X if every open set U containing x, we have

 \left\{ {A \cup U} \right\}\backslash \left\{ x \right\} = \phi

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 X = \left\{ {a,b,c} \right\} with topology \tau  = \left\{ {\phi ,\left\{ {a,b} \right\},\left\{ c \right\},X} \right\} andA = \left\{ a \right\}, then b is the only limit point of A, because the open sets containing b namely \left\{ {a,b} \right\} and X also contains a point a of A.

On the other hand, a and b are not limit point of C = \left\{ c \right\}, because the open set \left\{ {a,b} \right\} 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 \left\{ c \right\} containing c does not contain any other point of C different from c. Thus, the set C = \left\{ c \right\} has no limit points.

As another example, let X = \left\{ {a,b,c,d,e} \right\} with topology \tau  = \left\{ {\phi ,\left\{ a \right\},\left\{ {c,d} \right\},\left\{ {a,c,d} \right\},\left\{ {b,c,d,e} \right\},X} \right\}. Let A = \left\{ {a,b,c} \right\} then a is not a limit point of A, because the open set \left\{ a \right\} containing a does not contain any other point of A different from a. b is a limit point of A, because the open sets \left\{ {b,c,d,e} \right\} 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 \left( {X,\tau } \right) 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 D\left( A \right) or {A^d}. In the above example, D\left( A \right) = \left\{ {b,d,e} \right\}.

Remark:

It may be noted that under usual topology, consider the subsets \left[ {a,b} \right], \left( {a,b} \right), \left[ {a,b} \right), \left( {a,b} \right] of real, then all the points of these intervals are limits points.