Let $M$ be a smooth manifold and for $r, s\geq 0$ define the tensor bundle: $$T^{r, s}(M):=\bigcup_{p\in M} T_pM.$$ I'm trying to understand its topology. I'm following Homology and Curvature written by S. Goldberg. The construction shouldn't be so complicated. It goes as follows:
Let $(U, \phi)$ be a local chart of $M$ and write $\phi=(x_1, \ldots, x_n)$. Then it defines a basis $\{(\partial_i)_p\}_{i=1}^n$ for $T_pM$. Let $\{(dx_i)_p\}_{i=1}^n$ be the dual basis such that $$\{(\partial_{i_1})_p\otimes \ldots \otimes (\partial_{i_r})_p\otimes (dx_{j_1})_p\otimes \ldots (dx_{j_s})_p\}$$ is a basis for $T^{r, s}(T_pM)$ with $i_k, j_\ell$ varying all the possible ways.
Let $V$ be any $\mathbb R$-vector space of finite dimension $n$. We can define a map $$\varphi_{U}:U\times T^{r, s}(V)\longrightarrow T^{r, s}(M),$$ in a natural way assigning to each $(p, \omega)\in U\times T^{r, s}(V)$ the element of $T^{r, s}(T_pM)$ that has as components the same as $\omega$ once fixed a basis in $T^{r, s}(V)$.
The map $\varphi_U$ is $1-1$ and one can get a topology on $T^{r, s}(M)$ requiring that for each $U$, $\varphi_U$ maps open sets of $U\times T^{r, s}(V)$ into open sets of $T^{r, s}(M)$.
Question: I didn't understand the definition of this topology, can anyone write it explicitly?
Just to finish, each map $\varphi_U$ induces an injective map $$\varphi_{U, p}:T^{r, s}(V)\longrightarrow T^{r, s}(M), \omega\longmapsto \varphi_U(p, \omega).$$ If $V$ is another local chart of $M$ such that $U\cap V\neq \phi$ we define the map $$g_{UV}(p):=\phi^{-1}_{U, p}\circ \psi_{V, p}$$ which gives an element of $\textrm{Aut}(T^{r, s}(V))$ and we get the transition functions for the vector bundle $T^{r, s}(M)$.
Given any (real) vector space $V$ one can define a vector bundle on the topological variety $M$ as a space $E$ equipped with a projection map $$ \pi:E\longrightarrow M $$ satisfying the condition that for all $m\in M$ there is an open subset $m\in U\subset M$ such that $$ \pi^{-1}(U)\simeq U\times V $$ and $\pi$ restricted to $\pi^{-1}(U)$ is just projection onto the first factor. In other words, the topology in $E$ is such that locally $E$ is just a product space.
The case of the tensor bundle is just a special case of this more general situation and the paragraph you quote is just this idea made precise in that context.