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Home / Topology / Accumulation point| Topology

Accumulation point| Topology

Accumulation point: Let X be a topological space. A point pX 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 Aˊ, 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 bX 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 Aˊ = {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 En = {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 Ei where i ≤ p. If n0 ≤ 36 , then every open set containing n0 also contains 37 ∊ A which is different from n0; hence n0 ≤ 36 is a limit point of A. On the other hand, if n0 > 36 then the open set En0 = {n0, n0 + 1, n0 + 2, …} contains no point of A different from n0­. So n0 > 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}.

 

 

 

 

 

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