Exercise 3.5.9

Question

Decide whether the following sets are dense in $\mathbf{R}$, nowhere-dense in $\mathbf{R}$, or somewhere in between.
  \begin{enumerate}[label=(\alph*)] 
  \item $A=\mathbf{Q} \cap[0,5]$.
  \item $B=\{1 / n: n \in \mathbf{N}\}$.
  \item the set of irrationals.
  \item the Cantor set.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item between, since $A$ is dense in $[0,5]$ but not in all of $\mathbf R$.
  \item nowhere-dense since $\overline{B} = B \cup \{0\}$ contains no nonempty open intervals
  \item dense since $\overline{\mathbf I} = \mathbf R$
  \item nowhere-dense since the Cantor set is closed, so $\overline{C} = C$, and $C$ contains no intervals
   \end{enumerate}

Exercise 3.5.9

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Exercise 3.5.9

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