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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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The natural numbers (or even simpler sets) would be good for part a). For part b), you want every element of the set to be a limit point of the set. –  David Mitra Apr 22 '12 at 19:27
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You should specify the underlying space and the topology. I presume, from your speculation on the answer, that you are talking about $\Bbb R$ with the standard topology. Is this correct? –  David Mitra Apr 22 '12 at 19:35
    
I'm working on R –  Breton Apr 23 '12 at 2:27
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I'm not sure you know what a limit point 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. –  Gerry Myerson Apr 23 '12 at 3:34
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1 Answer

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.

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