Let $M$ be a manifold. I want to show that the tangent bundle $TM$ is second countable.
I know that for a given chart $(U, \phi)$ on $M$ we have a homeomorphism $D_{\phi}$ between $TU$ and $\phi(U) \times \mathbb{R}^m$. I also know that we can define a Topology on $TM$ via this map.
I want to construct a second countable basis for $TM$.
My attempt:
Let $E$ in $TM$ be open. By definition, this is the case iff for any chart $(U, \phi)$ of $M$, $D_{\phi} (E \cap TU)$ is open in $\phi(U) \times \mathbb{R}^m$. Since $\phi(U) \times \mathbb{R}^m$ is second countable it admits a countable Basis $B_{\phi}$. Then, since $D_{\phi}$ is a homeomorphism we have $E \cap TU= \bigcup_{B \in B_{\phi}, \ \ (D_{\phi})^{-1}(B) \subset E \cap TU} (D_{\phi})^{-1} (B)$. Now since $E$ is the union of $E \cap TU$ over all charts $(U,\phi)$ of $M$ I'm tempted to define the following basis of $TM$ : $B'= \{ (D_{\phi})^{-1} (B) | B \in B_{\phi}, \ \ (U, \phi)$ chart of $M \}$, wich i think, is countable since the set of charts on $M$ is countable ($M$ beeing second countable) and $B_{\phi}$ is countable, for each of those charts.
Does this make sense? Could you help me find a second countable basis?