How to show if there exists a global frame for a vector bundle, then the vector bundle is trivial.

Question

Let $\pi: E\to M$ be a vector bundle.

For each $p \in M$, if $(\sigma_i(p))$ is a global frame, then we can construct a map from $E \to M\times \mathbb R^k$ via $\Phi(\sigma_i(p))=(p, e_i)$. This map is bijective and linear. So, it is a vector space isomorphism from the fiber $\pi^{-1}(p) \to \{p\}\times \mathbb R^k$.

How can I show $\Phi$ a homeomorphism?