Remark

Let us retain the hypotheses and notations of proposition 8. Let G be a discrete topological group. Suppose that E is a principal covering of group G. If the coverings $E_t$ of B and $E_t\times T$ of $B\times T$ are endowed with the structures of principal coverings of group G deduced from that of E (I, p. 92, examples 1 and 4), then the coverings E and $E_t\times T$ are isomorphic principal coverings. Let in fact $h: E_t\times T\rightarrow E$ be the unique $(B\times T)$-morphism such that $h(x, t) =x$ for every $x\in E_t$. For every element $g$ of G, the mapping $(x, u)\mapsto h(x\cdot g, u)\cdot g^{-1}$ is a $(B\times T)$-morphism which maps $(x, t)$ to $x$ for every $x\in E_t$, hence is equal to $h$. This proves that $h$ is a $(B\times T)$-morphism of principal coverings.

In particular, under the preceding hypotheses, the principal coverings $E_t$ and $E_{t'}$ are isomorphic, for $t'\in T$. One proves analogously that if, in corollary 2, E and $E'$ are principal coverings of group G and if $k$ is a B-isomorphism of principal coverings of group G$,\widetilde{k}$ is a $(B\times T)$-isomorphism of principal coverings.