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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 ${e} \times G$.]

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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 probably necessary for multiplication on the right by a constant to be continuous though.]

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# Name for topologymaking group action followedbyquotientcontinuous

Given

Fix a topological space $(X, \tau)$ whose point set $X$ has a (not continuous) with right $G$-action by G$-action. Give$X$a topology$\tau$and make$G$a topological group$G$, we can form . (These topologies need not make the following construction: Consider action continuous). We can define another topology$G \times \tau'$on$X$with as the product largest topology making the action$G (X,\tau) \times \tau$. There is an equivalence relation$(x, g) G \sim to (y, h)$when$xg = yh$X,\tau')$ continuous. (This yields a is also called the quotient space topology on $G X$ with respect to the action $(X,\tau) \times X / G \sim$, whose point set to X$.) Note that if the$G$-action is identified with continuous for$X$via \tau$ then $x \tau'= \mapsto [(x, id)]$, but whose topology is not necessarily $\tau$.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?