My question is about the precise clarification (and source of proof) of the following statement I have heard of:
Let $H$ be an algebraic group action on a variety $X$. Let $x\in X$. Then $H.x$ is open in its Zariski closure.
I don't know what conditions I am supposed to impose on $H$ and $X$. But the scenarios I often work with is $H$ as a subgroup of $G$, acting on some $G/L$, where all groups here are algebraic over $\mathbb R$.
Can anyone tell me what the precise statement of this is and where to find a proof (please tell me the number of theorem you refer to in some book, thanks). I do not know schemes and stacks so please refrain from quotient theories that are too abstract for the above settings.