Fix a set $X$ with right $G$-action. Give $X$ a topology $\tau$ and make $G$ a topological group. (These topologies need not make the action continuous).
We can define another topology $\tau'$ on $X$ as the largest topology making the action $(X,\tau) \times G \to (X,\tau')$ continuous. (This is also called the quotient topology on $X$ with respect to the action $(X,\tau) \times G \to X$.)
Note that if the $G$-action is continuous for $\tau$ then $\tau'= \tau$.
For example, if $X = \mathbb{R}$, $\tau$ is the discrete topology and $G$ is $(\mathbb{R}, +)$ with the usual topology acting on $X$ by addition, then $G \times X / \sim = \mathbb{R}$ with the usual topology (unless I am much mistaken).
More interesting examples exist, e.g. the Skorokhod topology (again unless I am mistaken).
This construction feels useful enough that it must be well known and have a name. Can anyone provide me with more information?
[EDIT: actually I don't think it's necessary that $G$ is a topological group, just that it's a group with a topology. It's Although it is probably necessary for multiplication on inversion to be continuous at the right by a constant identity and for multiplication to be continuous though.on ${e} \times G$.]
[EDIT: made the presentation clearer to address the existing comments, changed title]
