I have a question about a condition in Mumford's GIT book.
One page 8, remark (6) iii) they talk about a closed subset $W$ of $X$ which is invariant under the action of $G$.
The context is that $G$ is a group in $Sch/S$ which acts on an $S$-scheme $X$. I do not understand what it means for a subset of $X$ to be invariant. The group action of $G$ in the category of schemes does not induce a group action on the underlying topological spaces and sets, so how does $G$ move the points of $|X|$ around?
In case that all schemes are of finite type over an algebraically closed field $k$, I would just look at the action of $G(k)$ on $X(k)$. But what does invariant under the action of $G$ mean in the general case?