Topological transformation group

Remarks

Example: the group $\mathrm{SO}(3)$ of rotations of $\R^3$, with the topology from the matrix entries, acting on the unit sphere $S^2$ by $(A, x) \mapsto Ax$. The map is given by polynomials in the entries of $A$ and the coordinates of $x$, so it is continuous. Non-example: $(\R, +)$ with the usual topology acting on $\R$ with the discrete topology by $(t, x) \mapsto t + x$. This is a group action, but it is not continuous: for fixed $x$, the map $t \mapsto t + x$ from $\R$ to the discrete space is not continuous. For each $g \in G$ the map $x \mapsto g \cdot x$ is a homeomorphism of $X$, with inverse $x \mapsto g^{-1} \cdot x$.