# When is a free and transitive $G$-space homeomorphic to $G$? And when is it (not) a torsor?

In response to comments: let $G$ be a topological group. A $G$-space is a topological space $X$ equipped with a continuous right group action: a continuous function $X \times G \to X$, $(x, s) \mapsto xs$, such that $x(st) = (xs)t$ and $x1 = x$.

Consider a right $G$-space $X$ such that the action of $G$ on $X$ is free and transitive. Generally speaking, $X$ is not homeomorphic to $G$, a counterexample is the action of $\mathbb{R}$ on $\hat{\mathbb{R}}$ by addition, where $\hat{\mathbb{R}}$ is $\mathbb{R}$ with left order topology.

I have two questions:
1) Are there broad sufficient conditions for when $X$ is homeomorphic to $G$? One example is when $X$ is Hausdorff and $G$ is compact (then $g \mapsto xg$ is the homeomorphism for any $x \in X$), are there any other?
2) For a torsor, by definition, $X \times G \to X \times X$, $(x, g) \mapsto (x, xg)$ is an isomorphism, and this implies that $X$ is homeomorphic to $G$. Why was this particular definition chosen for 'a group that forgot its unit'? Are there examples of free and transitive group actions where $X$ is homeomorphic to $G$, but is not a $G$-torsor? What's bad about them, what breaks down?

I have already asked for references in another question, but still if you know one that deals with these elementary issues, please point me to it :)

-
What is $G$? A topological group? How do you define a $G$-space, then? – Rasmus Sep 22 '11 at 12:54
Sure, $G$ is a topological group. – Alexei Averchenko Sep 22 '11 at 14:11
Which continuity condition do you want a $G$-action to fulfill? – Rasmus Sep 22 '11 at 14:57
The usual one: $X \times G \to X$ is continuous. – Alexei Averchenko Sep 22 '11 at 22:27