Example of subset of $\mathbb{R}²$ such that $A\neq A'\neq A''\neq A'''$?

Asked Jan 24 '15 at 19:25

Active Jan 24 '15 at 19:25

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I am looking for a subset of $\mathbb{R}²$ such that $A\neq A'\neq A''\neq A'''$ (where $A'$ is the set of limit points of $A$).

I read it's possible but I don't even see how it could be ... I've tried a subset that has an isolated point (or several) and is open otherwise.

The furthest I could go was picking $\lbrace(\frac{1}{n},0):n\in\mathbb{Z}\rbrace \cup B((2,0),1)$ because then $A'$ has an isolated point.

I'm sure I'm missing a point here, could you help me?

Thank you very much

asked Jan 24 '15 at 19:25

Noome

@Brian M. Scott, thank you! I did not see this one... – Noome Jan 24 '15 at 19:46
You’re welcome; there are some good answers there, so it should help. – Brian M. Scott Jan 24 '15 at 19:47
Yes, I had tried the solution the comments give, but got my computation wrong ^^ – Noome Jan 24 '15 at 19:49

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