Why is my proof that $\mathbb R$ is disconnected wrong?

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?

• Open sets. You're missing the point 'open sets'. – Anik Bhowmick Dec 8 '18 at 11:39
• Yes thank you, that would fix it – Toby Peterken Dec 8 '18 at 11:45
• You're welcome. – Anik Bhowmick Dec 8 '18 at 11:47

2 Answers

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.

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