Topological principal bundle

Let $G$ be a topological group and $X$ a topological space. A topological principal $G$ -bundle over $X$ consists of a topological space $P$ , a continuous surjection $\pi:P\to X$ , and a continuous right action $P\times G\to P$ , $(p,g)\mapsto pg$ , such that:

$pe=p$ , $(pg)h=p(gh)$ , and $\pi(pg)=\pi(p)$ .
There is an open cover $\{U_i\}$ of $X$ and homeomorphisms $\varphi_i:\pi^{-1}(U_i)\to U_i\times G$ over $U_i$ satisfying $\varphi_i(pg)=(x,hg)$ whenever $\varphi_i(p)=(x,h)$ .

The action is therefore free and transitive on each fiber. The topology on the local product is the product topology. A bundle isomorphism is a $G$ -equivariant homeomorphism over the base.

Pullback

For a continuous map $f:Y\to X$ , define $f^*P=\{(y,p):f(y)=\pi(p)\}$ with the subspace topology from $Y\times P$ , projection $(y,p)\mapsto y$ , and right action $(y,p)g=(y,pg)$ . Pulling back the local trivializations gives a topological principal $G$ -bundle over $Y$ .

Smooth bundles and universal bundles

A smooth principal bundle has an underlying topological principal bundle. The topological definition also applies when the base or total space is not a finite-dimensional manifold, as happens for universal bundles.