Example 4

Let B be a topological space, G a topological group, and let $(E, p)$ be a principal fibred B-space with group G. Let H be a subgroup of G.

Let us denote by $\varphi : E\rightarrow E\times (G/H)$ and $\psi : E\times (G/H)\rightarrow E/H$ the mappings defined by $\varphi (x) = (x,H)$ and $\psi (x, gH) = (x\cdot g)H$. They are compatible with the projections onto B and with the operations of G and define, by passing to quotients, morphisms of B-spaces $\overline{\varphi}: E/H\rightarrow E\times^G(G/H)$ and $\overline{\psi}: E\times^G(G/H)\rightarrow E/H$, inverse to one another. One says that $\overline{\varphi}$ is the canonical homeomorphism of $E/H$ onto $E\times^G(G/H)$. In particular, the topological space $E/H$ endowed with the continuous mapping $p_H: E/H\rightarrow B$ is a locally trivial fibred B-space of fibre type $G/H$.

If, moreover, H is a normal subgroup of G, the action of G endows the B-space $E/H$, by passing to quotients, with a structure of principal fibred space with group $G/H$.

In particular, if E is a principal covering of B with group G, $E/H$ is a covering of B; if H is normal in G, it is a principal covering with group $G/H$.