3
$\begingroup$

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!

$\endgroup$
  • 1
    $\begingroup$ $P$ and $Q$ are open in $A$ and $A=P\cup Q$. This is all, since $P\cap Q=\emptyset$. $\endgroup$ – egreg Jul 24 '13 at 9:08
  • $\begingroup$ Looks correct. Just make sure to mention that $P$ and $Q$ are open in the proof. $\endgroup$ – Ayman Hourieh Jul 24 '13 at 9:08
  • $\begingroup$ Great, thanks guys for your input. $\endgroup$ – Shant Danielian Jul 24 '13 at 9:15
1
$\begingroup$

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

(Posting this as a CW answer because the question would appear unanswered otherwise.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.