Theorem 3

Let B be a nonempty, connected and locally connected topological space. Every covering of B is associable to a Galois covering; every finite covering of B is associable to a finite Galois covering.

Let X be a covering of B, and let us denote by $q$ its projection. Since the space B is connected and nonempty, the covering X has a degree; let us denote this degree by F and endow the set F with the discrete topology. Since the space B is locally connected, the sheaf $\mathscr{I}$ = $\mathscr{I}$som$_B(B\times F; X)$ is locally constant and at every point $b$ of B its stalk $\mathscr{I}_b$ is canonically isomorphic to the set $\mathscr{B}(F; X_b)$ of bijections of F onto the fibre $X_b($I, p. 89, prop. 12). Let $E = E_{\mathscr{I}}$ be the covering associated with the locally constant sheaf $\mathscr{I}($I, p. 86, corollary).

Let G be the group of permutations of F. For every $g\in G$, one defines as follows a morphism $\gamma '(g)$ of the sheaf $\mathscr{I}$ into itself: for every open subset U of B, the mapping $\gamma '(g)_U$ associates with a U-isomorphism $\varphi : U\times F\rightarrow \overset{-1}{q}(U)$ the U-isomorphism defined by $(b, f)\mapsto \varphi (b, g(f))$ for $b\in U$ and $f\in F$. One has $\gamma '$(Id$_F$) $=$ Id$_{\mathscr{I}}$ and $\gamma '(g\circ g') =\gamma '(g')\circ \gamma '(g)$, for $g,g'\in G$, so that for every $g\in G,\gamma '(g)$ is an automorphism of the sheaf $\mathscr{I}$. The B-morphism $\gamma (g): E\rightarrow E$ associated with $\gamma '(g)$ is an automorphism (I, p. 50). If $x= [U, \varphi , b]$ is an element of E, where U is an open subset of B$,b$ a point of U and $\varphi$ a U-isomorphism of $U\times F$ onto $\overset{-1}{q}(U)$, one has $\gamma (g)(x) = [U, \psi , b]$, where $\psi$ is defined by $\psi (a, f) =\varphi (a, g(f))$, for $a\in$ U and $f\in$ F. If $g$ and $g'$ are elements of G, one has $\gamma (g\circ g') =\gamma (g')\circ \gamma (g)$. One has $\gamma$(Id$_F$) $=$ Id$_E$. One thus defines a continuous right operation law of G in E by setting $x\cdot g=\gamma (g)(x)$, for $x\in E$ and $g\in G$. The group G operates simply transitively on every fibre of E, so that the covering E endowed with this operation is a principal covering with group G (I, p. 97, prop. 4). Let $h$ be the mapping from $E\times F$ into X defined by $h([U, \varphi , b], f) =\varphi (b, f)$ for every open subset U of B, every point $b$ of U and every U-isomorphism $\varphi$ of $U\times F$ onto $\overset{-1}{q}(U)$. By definition of the topology of E, it is continuous; it is a B-morphism. For every element $g$ of G and every point $(x, f)$ of $E\times F$, one has $h(x, g(f)) =h(x\cdot g, f)$. The mapping $h$ therefore defines, by passing to the quotient, a B-morphism $h': E\times^GF\rightarrow X$. For every point $b$ of B, the mapping $h_b: E_b\times F\rightarrow X_b$ identifies with the mapping $(\varphi , f)\mapsto \varphi (f)$ from $\mathscr{B}(F; X_b)\times F$ into $X_b$, so that the mapping $h'_b$ is bijective. Consequently, $h'$ is a B-isomorphism of $E\times^GF$ onto X (I, p. 30, cor. 2 of prop. 6).

As the space B is connected, locally connected and nonempty, the covering X, associated with the principal covering E, is associated with a Galois covering (I, p. 110, corollary 2). If the covering X is finite, the same is true of the covering E, hence X is associated with a finite Galois covering (loc. cit.).