Proposition 1

Let $X, X', X''$ be three topological spaces, and let $f : X \to X', \ g : X' \to X''$ be two mappings. Then:

a) If $f$ and $g$ are open (resp. closed), so is $g \circ f$.

b) If $g \circ f$ is open (resp. closed) and if $f$ is continuous and surjective, then $g$ is open (resp. closed).

c) If $g \circ f$ is open (resp. closed) and if $g$ is continuous and injective, then $f$ is open (resp. closed).

From Definition 1. a) follows immediately. To prove b) it is enough to remark that every open (resp. closed) subset $A'$ of $X'$ can be written as $f(A)$, where $A = f^{-1}(A')$ is open (resp. closed) in $X$ (§ 2, no. 1, Theorem 1); hence $g(A') = g(f(A))$ is open (resp. closed) in $X''$. Finally, to prove c), we remark that $f(A) = g^{-1}(g(f(A)))$ for every subset $A$ of $X$; by hypothesis, if $A$ is open (resp. closed) in $X$, then $g(f(A))$ is open (resp. closed) in $X''$, hence $f(A)$ is open (resp. closed) in $X'$ by § 2, no. 1, Theorem 1.