Proposition 6

Let X, Y, Z be topological spaces and let $f: X\rightarrow Y,g: Y\rightarrow Z$ be mappings.

a) Suppose that $f$ and $g$ are etale; then $g\circ f$ is etale.

b) Suppose that $g$ is étale, $f$ is continuous and $g\circ f$ is open. Then $f$ is open.

c) Suppose that $g\circ f$ and $g$ are étale and that $f$ is continuous. The mapping $f$ is then étale.

d) Suppose that $g\circ f$ is étale and that the mapping $f$ is continuous and open; then $f$ is étale and $g$ is étale at every point of $f(X)$.

Let us prove a). Suppose the mappings $f$ and $g$ are étale. Then they are continuous and open, hence the mapping $g\circ f$ is continuous and open. Let $x$ be a point of X. There exists a neighbourhood W of $f(x)$ in Y such that the mapping $g|W$ is injective, and a neighbourhood V of $x$

in X, contained in $\overset{-1}{f}(W)$, such that the mapping $f|V$ is injective; the mapping $(g\circ f)|V$ is then injective. This proves that the mapping $g\circ f$ is étale (remark 2).

Let us prove b). Let $x$ be a point of X and W an open neighbourhood of $f(x)$ such that $g$ induces a homeomorphism of W onto the open set $g(W)$. Let V be a neighbourhood of $x$ such that $f(V)$ is contained in W; then $g\circ f(V)$ is a neighbourhood of $g\circ f(x)$, hence $f(V)$ is a neighbourhood of $f(x)$ in the open set W, and also in Y. This proves that the mapping $f$ is open (TG, I, p. 33, prop. 5).

Let us prove c). By b$),f$ is open. Let $x$ be a point of X. Since $g\circ f$ is étale, there exists a neighbourhood V of $x$ in X such that $g\circ f|V$ is injective. Thus, $f|V$ is injective and $f$ is étale (I, p. 29, remark 2).

Let us prove d). Let $x$ be a point of X and put $y=f(x)$. There exists an open neighbourhood V of $x$ such that $g\circ f$ induces a homeomorphism of V onto the open set $g\circ f(V)$. Since the mapping $f$ is open, $f(V)$ is an open neighbourhood of $y$. The mapping $f|V: V\rightarrow f(V)$ deduced from $f$ by passing to subspaces is continuous, open and bijective; hence it is a homeomorphism, so that $f$ is étale at $x$. Moreover, the mapping $g$ induces a homeomorphism of $f(V)$ onto $g\circ f(V)$, hence is étale at $y$.