Suppose that $ M $ is an $ n $-manifold and that $ (U,\phi) $ is a chart on $ M $, where $ U $ is an open subset of $ \mathbb{R}^{n} $. The claim is that $$ T U = \bigsqcup_{p \in U} T_{p} U = \bigsqcup_{p \in U} \mathbb{R}^{n} = U \times \mathbb{R}^{n}. $$
The identities actually mean that the objects are isomorphic as vector spaces. In this related thread, ptF’s answer says that $ W \subseteq T M $ is open if $ {d \phi}(W \cap T U) $ is open in $ \mathbb{R}^{2 n} $, where $ d \phi: T U \to T \mathbb{R}^{n} $ is the total derivative induced by $ \phi $.
Is this well-defined? $ {d \phi}(W \cap T U) $ is a subset of $ T \mathbb{R}^{n} $, on which a topology hasn’t been defined yet. I only know that $ T \mathbb{R}^{n} $ is isomorphic to $ \mathbb{R}^{n} \times \mathbb{R}^{n} $ as vector spaces, but can we say that something is open in $ T \mathbb{R}^{n} $?