The orbit map and the restriction of the orbit map both are open.

Question

Let $\phi: G \times X \to X$ (Often denoted by $(X,G)$) is a topological group action.

We know that for any $x\in X$, the orbit map $\phi_x:G \to X$ is a continuous mapping and the restriction $\varphi_x:G \to Gx$ is also a continuous mapping, where $Gx=\{gx:g \in G\}$.

I need an example about a dynamical system$(X,G)$, such that, the first, $\forall x\in X$, $\phi_x$ and $\varphi_x$ both are open mappings. Second, for any $x\in X$, $Gx\neq X$.

The following questions may help you: https://math.stackexchange.com/questions/1330967/in-which-condition-the-restriction-of-the-orbit-map-is-open

Restriction of the orbit map must be open?

Thanks a lot.

Answer 1

Take X be the disjoint union of two copies of G, and G acts by left translation on each copy