Exercise 3.2.2

Question

Let
  $$A=\left\{(-1)^{n}+\frac{2}{n}: n=1,2,3, \ldotsight\} \quad 	ext { and } \quad B=\{x \in \mathbf{Q}: 0<x<1\}$$
  Answer the following questions for each set:
  \begin{enumerate}[label=(\alph*)] 
  \item What are the limit points?
  \item Is the set open? Closed?
  \item Does the set contain any isolated points?
  \item Find the closure of the set.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item The set of $B$'s limit points is $[0, 1]$. The set of $A$'s limit points is $\{1, -1\}$.
  \item
    $B$ is not open since every $(a,b) 
ot\subseteq B$ and $B$ is not closed since we can construct limits to irrational values outside $B$.
    $A$ is closed since $\{1, -1\} \subseteq A$, but not open as it does not contain any irrationals meaning $(a,b) 
ot\subseteq A$ for all $a,b \in \mathbf{R}$.
  \item
    Every point of $A$ except the limit points $\{1, -1\}$ is isolated, as if it were not isolated it would be a limit point.
    $B$ has no isolated points since $B \setminus [0, 1] = \emptyset$, or in other words since $B$ is dense in $[0,1]$ every $b \in B \subseteq [0,1]$ can be reached via a limit.
  \item $\overline{enumerate}

Exercise 3.2.2

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Exercise 3.2.2

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