Definition 1

Let B be a topological space. Let E be a B-space and let $p$ be its projection. We say that E is a locally trivial fibred B-space if every point of B possesses a neighbourhood V such that $(\overset{-1}{p}(V), p_V)$ is a trivializable fibred V-space.

Instead of saying "locally trivial fibred B-space", we also say "locally trivial fibre space with base B". If E is a locally trivial fibred B-space, we sometimes say that $(E,B, p)$ is a locally trivial fibration, or, by abuse, that $p$ is a locally trivial fibration. We also say that a B-space $(E, p)$ is trivializable over a subset A of B if the A-space $E_A$ induced by $(E, p)$ over A is a trivializable fibre space.

Let E be a locally trivial fibred B-space; let us denote by $p$ its projection.

The set of points $a$ of B such that the fibre $\overset{-1}{p}(a)$ is empty (resp. nonempty) is open. The image of $p$ is therefore an open and closed subset of B.

Let F be a topological space. If all the fibres of E are homeomorphic to F, we say that E is a locally trivial fibred B-space of fibre type F.