principal bundle

Definition

Given topological spaces $P,X$ and a topological group $G$ that acts on $P$, a principal $G$-bundle is a fiber bundle

Such that the action of $G$ on $P$ preserves the fibers of $\pi$ and acts freely and properly such that the map

$$\begin{array}{rl}G& ⟶P_x\\ g& ⟼gp\forall p\in P_x\end{array}$$

is a homeomorphism.