Let X be a metric space and A⊂X then x∈A^{e} if and only if d(x,A)>0 where A^{e}=the set of points outside
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We prove the equivalent statement that d(x,A) = 0 if and only if x ∈ cl(A):
d(x,A) = 0 iff
for every r > 0 there exists a ∈ A such that d(x,a) < r iff
for every r > 0, B(x,r) ∩ A ≠ Ø iff
x ∈ cl(A).
d(x,A) = 0 iff
for every r > 0 there exists a ∈ A such that d(x,a) < r iff
for every r > 0, B(x,r) ∩ A ≠ Ø iff
x ∈ cl(A).