Exercise 3.4.7

Question

A set $E$ is totally disconnected if, given any two distinct points $x, y \in E$, there exist separated sets $A$ and $B$ with $x \in A, y \in B$, and $E=A \cup B$.
  \begin{enumerate}[label=(\alph*)] 
  \item Show that $\mathbf{Q}$ is totally disconnected.
  \item Is the set of irrational numbers totally disconnected?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Let $x,y \in \mathbf Q$, and let $z \in (x,y)$ with $z \in \mathbf I$. The sets $A = (-\infty, z) \cap \mathbf Q$ and $B = (z, \infty) \cap \mathbf Q$ are separated and have $A \cup B = \mathbf Q$.
  \item Now let $x,y \in \mathbf I$, and let $z \in (x,y)$ with $z \in \mathbf Q$. The sets $A = (-\infty, z) \cap \mathbf I$ and $B = (z, \infty) \cap \mathbf I$ are separated and have $A \cup B = \mathbf I$.
   \end{enumerate}

Exercise 3.4.7

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Exercise 3.4.7

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