Proposition 8

Let B be a topological space. Let T be a simply connected and locally connected space. Let E be a covering of $B\times T$, with projection $p$, and let $t$ be a point of T. Let $E_t$ denote the space $\overset{-1}{p}(B\times \{t\})$; endowed with the mapping $p_t=$ pr$_1\circ p|E_t: E_t\rightarrow B$, it is a covering of B. There then exists a unique $(B\times T)$-isomorphism of the covering $(E_t\times T, p_t\times$ Id$_T)$ onto the covering E which maps $(x, t)$ to $x$ for every $x\in E_t$.

One may suppose that B is not empty. Let $x$ be a point of $E_t$. By proposition 3 of I, p. 125 applied to the covering E and to the continuous mapping $T\rightarrow B\times T,u\mapsto (p_t(x), u)$, there exists a unique continuous mapping $f_x: T\rightarrow E$ such that $f_x(t) =x$ and $p(f_x(u)) =$ $(p_t(x), u)$ for every $u\in T$. Let $h: E_t\times T\rightarrow E$ be the mapping defined by $h(x, u) =f_x(u)$. We have $h(x, t) =x$ and $p\circ h=p_t\times$ Id$_T$. The mapping $h$ is a lifting to E of the mapping $p_t\times$ Id$_T$. The restriction of $h$ to $E_t\times \{t\}$ is continuous, as is also the restriction of $h$ to $\{x\} \times T$ for every point $x$ of $E_t$. Since the space T is locally connected, the mapping $h$ is continuous ( I, p. 37, cor. 1 of th. 1).

Let $b$ be a point of B. By construction, the mapping $h$ induces a bijection of the fibre $\overset{-1}{p_{t}}(b)\times \{t\}$ of $E_t\times T$ onto the fibre $\overset{-1}{p}(b, t)$ of E at $(b, t)$. Since the space T is connected and locally connected, the mapping $h$ is bijective (I, p. 84, cor. of prop. 7). It is therefore a $B\times T$-isomorphism (I, p. 30, cor. 2 of prop. 6).

Let $h'$ be a $(B\times T)$-isomorphism of the covering $(E_t\times T, p_t\times$ Id$_T)$ onto the covering E which maps $(x, t)$ to $x$ for every point $x\in E_t$. For every $x\in E_t$, the mappings $u\mapsto h(x, u)$ and $u\mapsto h'(x, u)$ are equal (I, p. 34, cor. 1 of prop. 11). Hence $h=h'$.