Example 1

Let $(E, p)$ be a principal fibre space with base B and group G, and let A be a subspace of B. The subspace $E_A=\overset{-1}{p}(A)$ of E is stable under the operation of G, and the mapping $p_A: E_A\rightarrow A$ makes it into a principal fibre space with base A and group G. One says that $(E_A, p_A)$ is the principal fibre space induced by $(E, p)$ over A.