Is a principal bundle over a discrete base space necessarily trivial?

Question

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?

Answer 1

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 $$(x,g) \mapsto (x, s(x) \cdot g)$$

Answer 2

Consider an extension of groups $1\rightarrow H\rightarrow G\rightarrow K\rightarrow 1$. where $H,G,K$ are finite. It is a principal bundle over $K$ with typical fiber $H$. If the extension is not trivial the bundle is not trivial.