Exercise 3.2.10

Question

Only one of the following three descriptions can be realized. Provide an example that illustrates the viable description, and explain why the other two cannot exist.
  \begin{enumerate}[label=(\roman*)] 
  \item A countable set contained in $[0,1]$ with no limit points.
  \item A countable set contained in $[0,1]$ with no isolated points.
  \item A set with an uncountable number of isolated points.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\roman*)] 
  \item Cannot exist because taking any sequence $(x_n)$ BW tells us there exists a convergent subsequence.
  \item $\mathbf Q \cap [0,1]$ is countable and has no isolated points.
  \item Impossible, let $A \subseteq \mathbf{R}$ and let $x$ be an isolated point of $A$. From the definition there exists a $\delta>0$ with $V_\delta(x) \cap A = \{x\}$. in Exercise 1.5.6 we proved there cannot exist an uncountable collection of disjoint open intervals, meaning we cannot have an uncountable set of isolated points as we can map them to open sets in a 1-1 fashion.
   \end{enumerate}

Exercise 3.2.10

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Exercise 3.2.10

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