When a free action gives rise to a $G$-principal bundle
Let a (topological) group $G$ act freely on a (topological) space $X$. Assume that
$G \backslash X$ is Hausdorff. (equivalently the image of the map $G \times X \to X\times X$ is closed).
Does it mean that $X \to G \backslash X$ is a $G$-principal bundle?
We are interested in the case when both $G$ and $X$ are l-spaces and moreover when $G=\mathbb Z^n$. However, I'll be happy to hear about other contexts too: locally compact spaces, topological manifolds, $C^\infty$-manifolds, algebraic varieties, etc.