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Give an example of a set which

$\ \ \ $a) contains a point which is not a limit point of the set

$\ \ \ $b) contains no point which is not a limit point of the set

In part b), I think it might be the naturals...

Could someone help me through this problem?

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    I'm not sure you know what a limit $p$oint is. Perhaps you should make sure you understand the definition. If you are sure you understand the definition, but can't do the problems, post the definition (edit it into your question), and someone will help you get from the definition to the answers.2012-04-23

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Assuming that your definition of limit point is the following:

(Def) A limit point $x$ of a set $A$ in a topological space $X$ is a point such that for every open set $O$ such that $x \in O$: $(O \setminus \{x\}) \cap A \neq \varnothing$.

In $\mathbb R$ with the standard metric this means that for $a)$ you want a set of isolated points, for example $\{0\}$.

For $b)$ any open set will do, for example $(0,1)$ doesn't contain any non-limit points.