I've been trying to read a little bit about vector bundles and one of the first examples one finds is the real line bundle over $\mathbb{R}P^n$ and more generically these $k$ bundles over the Grassmannians. However the topology over the total space $E$ is left undefined where $$E = \{(W, \omega) : W \in Gr(\mathbb{R}^n), \omega \in W\}$$
Are we just identifying each equivalence class with a vector space already and taking the product topology?
Note that $E = \{(W, \omega) \in Gr(\mathbb{R}^n)\times\mathbb{R}^n \mid \omega \in W\} \subset Gr(\mathbb{R}^n)\times\mathbb{R}^n$. The topology on $E$ is the subspace topology.