# Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle?

Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\mathbb{R}$.

What about if we change the topology by consideration of $p-$adic topology on rational numbers?

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A space is called ultraparacompact if every open cover can be refined to a cover by disjoint open sets. Note that clearly any fiber bundle on an ultraparacompact space is trivial (just apply the definition to an open cover on which it is trivial). So it suffices to show that $\mathbb{Q}P^n$ is ultraparacompact (with either the standard topology or the $p$-adic topology). More generally, I will prove the following:

Theorem: Any countable space with a basis of clopen sets is ultraparacompact.

Proof: Let $Q=\{q_n\}$ be a countable space with a basis of clopen sets and let $\mathcal{U}$ be an open cover. Define a refinement of $\mathcal{U}$ by induction. First, let $V_1$ be any clopen set containing $q_1$ which is contained in some element of $\mathcal{U}$. Next, if $q_2\in V_1$, let $V_2=V_1$; otherwise, let $V_2$ be a clopen set containing $q_2$ which is contained in some element of $\mathcal{U}$ and disjoint from $V_1$. Continue by induction, adding clopen sets contained in some element of $\mathcal{U}$, containing the next $q_n$, and disjoint from the clopen sets we’ve chosen so far. In the end, we get an open cover $\{V_n\}$ refining $\mathcal{U}$ consisting of disjoint clopen sets.