Adherent and limit point

Adherent point:

Def. A point p ∈ R is said to be an adherent point of a set A ⊂ R if every neighbourhood of p contain of A. The set of all adherent points of A is called the adherence of A and is denoted by Adh A.

If ε > 0

(p – ε, p + ε) ∩ A ≠ ∅

Exp –

Let A = {1,2,3,4,5}

p = 3

take ε> 0­­­­­­­­­­

(3- ε, 3+ ε)­­­ ∩ A ≠ ∅

(3- ε, 3+ ε)­­­ ∩ A = 3

Exp- ­­­

Set a A set (0,1)­­­­­­­­­

Let p = 0

ε> 0­­­­­­­­­­

(0-ε,0 + ε) ∩ A ≠ ∅

(0-ε,0 + ε) ∩ A = 0 + ε

Its means if point does not member of a set than it is possible this point is adherent point of a set

Limit Point

A point p ∈ R is said to be a limit point of a set A ⊂ R if every neighbourhood of contains­­­­­­ a point of A distinct form p.

(p – ε, p + ε) ∩ A – {p} ≠ ∅­­

Exp –

Let a A set (-1, 1) , A ⊂ R

Now let p = 0

ε> 0

(0 – ε, 0 + ε) ∩ A – {0} ≠ ∅­­

OR

(0 – ε, 0 + ε) ∩ (-1,1) – {0} ≠ ∅­­