i have a problem about the definition of isolated point. In my notes, it says that a boundary point z of A may not be an accumulation point. To prove this, it says if z is an isolated point of A, then there is a ball B(z,r) such that B(z,r) intersection with A = {z} and this z is the isolated point. The only possibility i think is that if A is a discrete metrics then this situation is possible. Otherwise, that ball includes infinitely many points. Am i right here? What exactly is an isolated point? I would appreciate if someone could help. Thanks
Isolated point in metric spaces
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metric-spaces