**Accumulation point**: Let* X *be a topological space. A point *p* ∊ *X* is an accumulation point or limit point (also called cluster point or derived point) of a subset *A *of *X* if and only if every open set *G *containing *p* contains a point *A* different from *p*, i.e.,

*G* open, *p* ∊* G *implies ( *G* \ {*p*}) ⋂ *A* ≠ ∅

The set of accumulation points of *A*, denoted by* A*^{ˊ}, is called the derived set of *A.*

**Or**, a point *p* is a limit point of the set *E* if every neighborhood of *p* contains a point *q *≠ *p* such that *q* ∊* E*.

**Example**: The class T = {*X*, ∅, {*a*}, {*c, d*}, {*a, c, d*}, {*b, c, d, e*}} defines a topology on *X* = {*a, b, c, d, e*}. Consider the subset *A *= {*a, b, c*} of *X*. Observe that *b* ∊ *X* is a limit point of *A* since the open sets containing b are {*b, c d, e*} and *X*, and each contains a point of *A *different from b, i.e. *c*. On the other hand, the point *a *∊ *X *is not a limit point of *A* since the open set {*a*} which contains {*a*} does not contains *a*, does not contain a point of *A* different from *a*. Similarly, the points *d* and *e* are limit points of *A* and the point *c* is not a limit point of *A*.

So *A*ˊ = {*b, d, e*} is the derived set of *A*.

**Problem**: Let T be the topology on N which consists of ∅ and all subsets of **N** of the form E_{n} = {n, n+1, n + 2, …} where n ∊ **N**. Find the accumulation points of the set A = {4, 13, 28, 37}

**Solution**: Observe that the open sets containing any point p ∊ **N** are the sets E_{i} where *i* ≤ p. If n_{0} ≤ 36 , then every open set containing n_{0} also contains 37 ∊ A which is different from n_{0}; hence n_{0} ≤ 36 is a limit point of A. On the other hand, if n_{0} > 36 then the open set E_{n0} = {n_{0}, n_{0} + 1, n_{0} + 2, …} contains no point of A different from n_{0}. So n_{0} > 36 is not a limit point of A. Accordingly, the derived set of A is Aˊ = {1, 2, 3, …, 34, 35, 36}.

**Problem**: Consider the topology

T = {*X*, ∅, {*a*},{*a, b*}, {*a, c, d*}, {*a*, *b, c, d*}, {*a, b, e*}}

On X = {*a, b, c, d, e*}. Determine the derived set of *A* = {*c, d, e*}

**Solution**: Observe that {a, b} and {a, b, e} are open subsets of X and that

*a, b *∊ {*a, b*} and {*a, b*} ⋂ *A* = ∅

*e *∊ {*a, b, e*} and {*a, b, e*} ⋂ *A* = {*e*}

Hence a, b and e are not limit points of A. On the other hand, every other point in X is a limit point of A since every open set containing it, also contains a point of A different from it. Accordingly, Aˊ = {c, d}.