On the real line, prove that the set of nonzero real numbers is not a connected set.

I'm studying Intro to Topology by Mendelson.

The problem is stated in the title.

My proof is,

Let $A=\mathbb{R}-\{0\}$. Then $C(A)=\{0\}$. Moreover, let $P=(0,\infty)$ and $Q=(-\infty,0)$. Then $A\subset P\cup Q$ and $P\cap Q=\emptyset\subset C(A)$. Also, $P\cap A\neq\emptyset$ and $Q\cap A\neq\emptyset$. Hence, $A$ is disconnected.

I feel like once I broke up $A$ into a union of disjoint open sets in $\mathbb{R}$ that $A$ is disconnected, but would this be enough for a proof? I went ahead and followed a theorem showing how to prove a set is disconnected, just to familiarize myself with the theorem, yet I feel it's unnecessary.

Thanks for any feedback!

• $P$ and $Q$ are open in $A$ and $A=P\cup Q$. This is all, since $P\cap Q=\emptyset$. – egreg Jul 24 '13 at 9:08
• Looks correct. Just make sure to mention that $P$ and $Q$ are open in the proof. – Ayman Hourieh Jul 24 '13 at 9:08
• Great, thanks guys for your input. – Shant Danielian Jul 24 '13 at 9:15

Your proof is correct. Just make sure to mention that $P$ and $Q$ are open in the proof.