Bundle - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A bundle in the broad topological sense used here is a triple $(E,\pi,B)$ , where $E$ and $B$ are topological spaces and $\pi:E\to B$ is a continuous surjection. The space $E$ is the total space, $B$ is the base space, and $\pi$ is the bundle projection. Its fiber over $b\in B$ is the subspace

E_b=\pi^{-1}(b)=\{e\in E:\pi(e)=b\}.

This definition alone imposes no local product condition.

Local triviality and terminology

A fiber bundle adds local triviality with a fixed model fiber. A smooth fiber bundle requires smooth local product charts. Many authors use “bundle” as shorthand for one of these stronger notions; the surrounding category and adjectives specify the intended meaning.

Example without local triviality

The map $\pi:\mathbb R\to[0,\infty)$ , $\pi(t)=t^2$ , is a continuous surjection. Its fiber over $0$ has one point, whereas each positive fiber has two. Thus it is a bundle in the broad sense above, but no neighborhood of $0$ admits a trivialization with one fixed fiber.