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The definition of connectedness in my notes is: A topological space $X$ is connected if there does not exist a pair of non empty subsets $U$, $V$ such that $U\cap V=\emptyset$ and $U\cup V=X$.

However if I have the subsets $(-\infty,0]$ and $(0,\infty)$ then these are disjoint and cover $\mathbb R$ and hence $\mathbb R$ is disconnected.

However $\mathbb R$ is clearly connected. Where have I gone wrong?

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    $\begingroup$ Open sets. You're missing the point 'open sets'. $\endgroup$ – Anik Bhowmick Dec 8 '18 at 11:39
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    $\begingroup$ Yes thank you, that would fix it $\endgroup$ – Toby Peterken Dec 8 '18 at 11:45
  • $\begingroup$ You're welcome. $\endgroup$ – Anik Bhowmick Dec 8 '18 at 11:47
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With your definition, every space $X$ with at least two points would be disconnected: just take a point $x\in X$ and consider $X=\{x\}\cup(X\setminus\{x\})$.

The definition requires $U$ and $V$ to be disjoint nonempty open sets such that $U\cup V=X$.

The set $(-\infty,0]$ is not open.

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The subsets that you take are wrong because $(-\infty ,0]$ contains a accumulation point of $(0,\infty)$.

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