In A Comprehensive Introduction to Differential Geometry by Spivak, on page 71, we have the following. In this context, $M$ is an abstract smooth manifold.
For any imbedding $i:M\to\Bbb R^N$, however, the structure of $T(M,i)$ is always simple locally: if $(x,U)$ is a coordinate system on $M$, then $\pi^{-1}(U)$, the part of $T(M,i)$ over $U$, can always be mapped, fibre by fibre, homeomorphically onto $U\times \Bbb R^n$. In fact, for each $p\in U$, the fibre $$ (M,i)_p \quad \text{equals}\quad (i\circ x^{-1})_{*x(p)}\big(\Bbb R^n_{x(p)}\big) = m_p\big(\Bbb R^n_{x(p)}\big), $$ where the abbreviation $m_p$ has been introduced temporarily; we can therefore define $$ f:\pi^{-1}(U)\to U\times\Bbb R^n $$ by $$ f\big(m_p(v_{x(p)})\big) = (p,v). $$ In standard jargon, $T(M,i)$ is "locally trivial."
My question is, how can I tell that $f$ is a homeomorphism like Spivak claims it is? I am unsure what the topology on $T(M,i)$ is, so I can't evaluate the continuity of this map.