Definition 5

Let B be a topological space, let G be a discrete topological group and let E be a principal covering of B with group G. A covering X of B is said to be associated with the principal covering E if there exists a G-set F and a B-isomorphism of $E\times^GF$ onto X.

Let B be a topological space, let G be a discrete topological group and let E be a covering of B, principal with group G.

For every covering X of B, the group G operates on the left on $\mathscr{C}_B(E; X)$ by the law defined by $(g\cdot h)(x) =h(x\cdot g)$ for $x\in E$, $g\in G,h\in \mathscr{C}_B(E; X)$.

For every G-set F endowed with the discrete topology, let us define a mapping $\alpha_F: F\rightarrow \mathscr{C}_B(E; E\times^GF)$ by:

(2) $\alpha_F(y)(x) =\pi (x, y)$ for $y\in F$ and $x\in E$,

where $\pi : E\times F\rightarrow E\times^GF$ is the canonical surjection. The mapping $\alpha_F$ is compatible with the operations of G on F and on $\mathscr{C}_B(E; E\times^GF)$.

For every covering X of B, there exists a unique mapping $\beta_X$ from $E\times^G\mathscr{C}_B(E; X)$ into X such that:

(3) $\beta_X(\pi (x, h)) =h(x)$ for $h\in \mathscr{C}_B(E; X)$ and $x\in E$.

In fact, one has $(g^{-1}\cdot h)(x\cdot g) =h(x)$ for $x\in E,g\in G$ and $h\in \mathscr{C}_B(E; X)$, by definition of the operation of G on $\mathscr{C}_B(E; X)$. When the set $\mathscr{C}_B(E; X)$ is endowed with the discrete topology, the mapping $\beta_X$ is a B-morphism of coverings.

When $X = E\times^GF$, one has

(4) $\beta_X(\pi (x, \alpha_F(y))) =\pi (x, y)$ for $x\in X$ and $y\in F$.