Let $X = [0,1] \cup [2,3]$ be a metric space with the euclidean metric, is it connected?
Im not sure because i think..
- It is connected because $X$ is complete and clearly not path connected.
- It is incomplete because $X$ cannot be written as the union of 2 non empty disjoint open sets. They can only be written as the union of 2 non empty disjoint closed sets $[0,1]$, $[2,3]$.