The irrational rotation on the torus $\mathbb{T^2}$: $T_t (z,w) = (e^{2 \pi i \alpha }z, e^{2 \pi i \beta} w)$, where $\alpha, \beta \in \mathbb{R}, \frac{\alpha}{\beta} \notin \mathbb{Q}$. It is perhaps the first nontrivial example of an ergodic dynamical system. By considering its orbits $t \mapsto T_t (z,w)$, one gets examples of one-to-one immersions (with dense image) which are not embeddings, immersed submanifolds which are not closed/regular submanifolds, Lie subgroups which are not closed subgroups, non-Hausdorff quotient space and similar phenomena.