Two different dense open subsets

Question

Could anybody give an example of two different subsets of space $(X,d)$ which are open and dense?

Thanks for help!

Answer 1

In $\mathbb{R}$ in the usual topology, $\mathbb{R}\setminus \mathbb{Z}$ is open and dense, and so are all sets of the form $\mathbb{R}\setminus F$ where $F$ is finite (e.g. a singleton). $\mathbb{R}\setminus C$ where $C$ is the middle third Cantor set is another still, as is $\mathbb{R}\setminus \{x: \exists n \in \mathbb{N}^+: x=\frac{1}{n} \lor x=0\}$ etc.

Plenty of open dense sets in most metric spaces. Only in the discrete metric/topology there is only one open dense subset, namely the whole space.