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

Question

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?

Answer 1

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

Answer 2

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.