Exercise 2.4.1

Question

Let $\left(X, d_{	ext {disc }}ight)$ be a metric space with the discrete metric. Let $E$ be a subset of $X$ which contains at least two elements. Show that $E$ is disconnected.

Prakash Anand · Accepted Answer

\begin{proof}
Let $x, y \in E$ and consider $V=\{x\}, W=E \backslash\{x\}$. So we have $V \cup W=E, V \cap W=\emptyset$, and $W 
eq \emptyset$ as $y \in W$, now we just need $V, W$ to be open in $E$. But this is clear since $V=\{x\}=B_{	ext {disc }}(x, 1 / 2)$ and $W=E \backslash\{x\}$ (since $\{x\}$ is closed, and so the complement is open) which are both open in $E$. Hence $E$ is disconnected.
\end{proof}

Exercise 2.4.1

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Exercise 2.4.1

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