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.
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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
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Exp-
Set a A set (0,1)
Let p = 0
ε> 0
(0-ε,0 + ε) ∩ A ≠ ∅
(0-ε,0 + ε) ∩ A = 0 + ε
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Its means if point does not member of a set than it is possible this point is adherent point of a set
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Limit Point
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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.
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(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} ≠ ∅