Isolated point

[ang3l]

Let S be a subest of R^N.

A point x in S is called an isolated point of S if there exists r >0 such that B_r(x) intersection S = {x} (that is x is the only point in S in the r-neighbourhood of x).

Prove that, if every point in S is isolated, then the closure of S has an empty interior: int(cl(S)) = empty set.

Thanks!!! :confused:

[OfficeShredder]

What is the closure of S in this specific case?

[ang3l]

I typed the question word by word from an assignment question that was given. He doesnt really specifiy sorry

[OfficeShredder]

I typed the question word by word from an assignment question that was given. He doesnt really specifiy sorry

No, I meant can you figure out what the closure of S is?

[HallsofIvy]

A related question, ang3l, is "If S consists only of isolated points, what is the interior of S itself?". You will see why that is important after you have answered Office Shredder's question.