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Here is my proof of that the cofinite topology on an infinite set $X$ is connected.

$X$ is connected $\iff$ There are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$.

Let $U, V \in X$ be non-empty disjoint open subsets of $X$ such that $U \cup V = X$.

As $U \cup V = X$ and $U \cap V = \emptyset$,

$\implies U^c = V$ and $V^c = U$.

$\implies U$ and $V$ are finite sets.

$\implies U \cup V$ is finite.

But $U \cup V = X$ which is infinite so we have a contradiction. Hence there are no non-empty disjoint open subsets $U, V \subseteq X$ such that $U \cup V = X$. So $X$ is connected.

Is my understanding correct?

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    $\begingroup$ Yes, it's correct. $\endgroup$ – Daniel Fischer Apr 1 '14 at 12:16
  • $\begingroup$ {deleted}...cheers. $\endgroup$ – sonicboom Apr 1 '14 at 12:57
  • $\begingroup$ That is "if and only if". It's often the definition of connectedness. $\endgroup$ – Daniel Fischer Apr 1 '14 at 13:01
  • $\begingroup$ Who bunped this question...? $\endgroup$ – IAmNoOne Nov 7 '14 at 1:56
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(To remove this from the unanswered list)

Your proof is correct.

There are not even disjoint non-empty open sets in this space.

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