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What steps should one take in order to find out the closure of a subset in a topological space? For example given $(\mathbb{R}, \tau_{0})$ as the topological space and the subset $$A = \left \{ \frac{1 + (-1)^n}{2} + (-1)^n \frac{n - 2}{3n - 2}, n \epsilon \mathbb{N} \right \}.$$ How can one find the closure of $A$?

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What is $\tau_0$? –  Chris Eagle Nov 9 '12 at 17:22
    
$\tau_{0}$ is the usual topology on $\mathbb{R}$. –  Mihai Bişog Nov 9 '12 at 17:25
    
It would have been clearer to just say "given $\Bbb R$ as the topological space". –  MJD Nov 9 '12 at 19:48

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up vote 4 down vote accepted

In general it depends entirely on the particular space and particular subset. Here the first step is to see just what $A$ looks like, where

$$A=\left\{\frac{1 + (-1)^n}{2} + (-1)^n \frac{n - 2}{3n - 2}:n\in\Bbb N\right\}\;.\tag{1}$$

It’s convenient to split $A$ into two subsets, one for even $n$ and one for odd $n$:

$$A_e=\left\{1+\frac{n-2}{3n-2}:n\in\Bbb N\text{ and }n\text{ is even}\right\}\;,$$ and

$$A_o=\left\{-\frac{n-2}{3n-2}:n\in\Bbb N\text{ and }n\text{ is odd}\right\}\;.$$

The sets $A_e$ and $A_o$ are actually the ranges of simple convergent sequences. What are $$\lim_{n\to\infty}\left(1+\frac{n-2}{3n-2}\right)\quad\text{ and }\quad\lim_{n\to\infty}\left(-\frac{n-2}{3n-2}\right)\;?$$ Can you see that these limits are limit points of $A_e$ and $A_o$, respectively?

Now make a sketch and decide whether $A$ has any other limit points. If so, identify them and explain why they are limit points; if not, try to show why no other point of $\Bbb R$ is a limit point of $A$.

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I think I understood it, give me some time to pice everything together. –  Mihai Bişog Nov 9 '12 at 17:57
    
@Thomas: I forgot to add the qualifiers after the colon. (I didn’t want to make the substitutions $n=2m$ and $n=2m+1$.) –  Brian M. Scott Nov 9 '12 at 18:24

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