I've seen "Theorem 2" in Rosenlicht's "Some Basic Theorems on Algebraic Groups" (http://www.jstor.org/stable/2372523) in different forms in varying texts. I'm having some trouble with the language in the aforementioned paper.
As I understand it, Rosenlicht's Theorem says that for an algebraic group $G$ over a field $k$ acting on an irreducible variety $X$ over $k$, there exists a Zariski-open, $G$-invariant, subset $X_0\subset X$ such that the geometric quotient exists. The geometric quotient is defined as a morphism $\pi:X_0\rightarrow Y$ satisfying
(1) $\pi$ is surjective, and for $x\in X_0$, $\pi^{-1}(\pi(x))$ equals the orbit of $x$ under the action of $G$.
(2) $\pi$ is an open map.
(3) For any open subset $U\subset Y$, the ring homomorphism $\pi^*$ is an isomorphism from $k[U]$ to $k[\pi^{-1}(U)]$.
In particular this implies that the orbits of $G$ on $X_0$ are closed. Is this correct? More specifically, do we need the condition that $k$ is algebraically closed? Can $k=\mathbb{R}$?