What are the examples of open sets that are compact?
Is the entire space X compact?
It depends on what topology you give on the space X.
For example, if you give a discrete topology on a finite set X, then every open set is compact.
Another example is that every open set is compact in the finite complement topology on $\displaystyle \mathbb{Re}$.
Meanwhile, if you give a usual topology on $\displaystyle \mathbb{Re}$, then an open compact set is an empty set only.
If you give a subspace topology on A with respect to the usual topology on $\displaystyle \mathbb{Re}$, where A is a finite subset of $\displaystyle \mathbb{Re}$, then every open set in the topological space A is compact.