nLab trivial vector bundle

Idea

For $X$ a suitable space then a vector bundle over $X$ is called trivial if there is no twist in how the fibers vary over it.

Definition

For $X$ a topological space, then a topological vector bundle $E \to X$ over a topological field $k$ is called trivial if its total space is the product topological space

$$E = X \times k^n \overset{pr_1}{\longrightarrow} X$$

with the topological vector space $k^n$, for some $n \in \mathbb{N}$. For $n = 1$, one also speaks of a trivial line bundle.

An isomorphism of vector bundles over $X$ of the form

$$E \longrightarrow X \times \mathbb{R}^n$$

is called a trivialization of $E$. If $E$ admits such an isomorphism, then it is called a trivializable vector bundle.