TITLE: Is a principal bundle over a discrete base space necessarily trivial?
QUESTION [5 upvotes]: Let $X$ be a topological space with discrete topology. More concretely, one can think of the lattice $\mathbb{Z}^n$.
Then, is it true that any principal $G-$bundle over $X$ is trivial? Here $G$ is an arbitrary Lie group.
I feel this way because every one-point set is open in $X$ but cannot prove rigorously myself.. Could anyone please help me?

REPLY [5 votes]: A bundle is trivial if and only if it has a global section. Take any set theoretic section of the bundle. Since the base is discrete, it will be continuous. So we have
$$s\colon X \to P$$
continuous, such that
$$p\circ s = \mathbb{1}_X$$
The isomorphism of the trivial bundle with the our bundle is given by the map
$$(x,g) \mapsto (x, s(x) \cdot g)$$