Let $n\in\mathbb N$, let $E,B$ be topological spaces and let $p:E\to B$ be a continuous map. For every $b\in B$, let $p^{-1}(b)$ be equipped with the structure of an $n$-dimensional real vector space. Regard the two following definitions of a vector bundle:
$p:E\to B$ is called a vector bundle if for every $b\in B$ there is an open neighbourhood $U\subset B$ of $b$ and a homoeomorphism $\phi: p^{-1}(U)\to U\times\mathbb R^n$ such that $\pi_1\circ\phi=p|_{p^{-1}(U)}$, and for all $y\in U$ the map $\pi_2\circ\phi: p^{-1}(y)\to\mathbb R^n$ is linear.
$p:E\to B$ is called a vector bundle if it is a fibre bundle with fibre $\mathbb R^n$ and the maps $$+:E\times_B E\to E, (x,y)\mapsto x+y$$ $$\cdot:\mathbb R\times E\to E, (t,x)\mapsto tx$$ are continuous.
Are these definitions equivalent?
If not, are the analogues in the category of smooth manifolds equivalent?
Edit: Removed a superfluous condition in the second definition.