Example of non-discrete space that admits an infinite open cover formed by pairwise disjoint open spaces [closed]
Can someone give me an example of a non-discrete space that admits an infinite open cover formed by pairwise disjoint open spaces? I'm trying to find some but I can't. I would appreciate some help.
Solution 1:
$\mathbb R - \mathbb Z = \bigcup_{n \in \mathbb Z} \, (n-1,n)$.
The space $\mathbb R - \mathbb Z$ is not discrete, each set $(n-1,n)$ is open in $\mathbb R - \mathbb Z$, and if $m \ne n \in \mathbb Z$ then the two sets $(m-1,m)$ and $(n-1,n)$ are disjoint.