In the introductory book by Milnor and Stasheff on characteristic classes, to define the Chern classes of some vector bundle $\varphi : E \to B$, we construct a vector bundle over the total deleted space $E_0$ of $\varphi$, where the total deleted space is the subspace of non-zero vectors in $E$. However, to have good properties, we frequently need to restrict to paracompact spaces. So it made me wonder: is the total deleted space paracompact? This is not discussed in the book.
The restriction of $\varphi$ to $E_0$ gives a $(\mathbb{R}^n \setminus \{0\})$-bundle over $B$, so we could generalize the question by asking if the total space of a $(\mathbb{R}^n \setminus \{0\})$-bundle over a paracompact Hausdorff space is paracompact. It seems likely to be true: by a result of Suzuki stated in Product of paracompact spaces, for any paracompact Hausdorff space $X$, we have that $X \times (\mathbb{R}^n \setminus \{0\})$ is paracompact. However, an open subset of a paracompact space is not paracompact in general, so it is not easy to work locally. I should probably use the fact that a paracompact Hausdorff space is normal, but I'm not sure how.
I would be interested to have a direct proof or a reference. The question Paracompactness of the projectified bundle over a paracompact space seems really similar, but I'm not sure if it answers my question or not.
Let $\varphi : E \to B$ be an $(\mathbb{R}^n \setminus \{0\})$-bundle, with $B$ Hausdorff paracompact, we will show that $E$ is paracompact. We can use the fact that the image of a paracompact space by a closed continuous map is paracompact (proof here by E. Michael). Let $(U_i)_{i \in I}$ be a collection of open subsets covering $B$ over which $\varphi$ is trivial, and which is locally finite. Then, we can find a collection $(V_i)_{i \in I}$ over the same index set $I$ with the same properties, and such that $\overline{V_i} \subseteq U_i$. Each $\overline{V_i}$ is paracompact, so $\overline{V_i} \times (\mathbb{R}^n \setminus \{0\})$ is paracompact, and therefore $\varphi^{-1}(\overline{V_i})$ is paracompact. The map $$ \bigsqcup_{i \in I} \varphi^{-1}(\overline{V_i}) \to E $$ is surjective, and closed because the collection is locally finite. Thus, $E$ is paracompact.