Theorem 1   The composition of two continuous mappings (when defined) is continuous.

Proof. Let $X,Y,Z$ be topological space, and let $f,g$ be mappings \begin{eqnarray*} f\colon X&\to& Y, \\ g\colon Y&\to& Z. \end{eqnarray*}We wish to prove that $g\circ f$ is continuous. Suppose $B$ is an open set in $Z$ . Since $g$ is continuous, $g^{-1}(B)$ is an open set in $Y$ , and since $f$ is continuous, $f^{-1}(g^{-1}(B))$ is an open set in $X$ . Since $f^{-1}(g^{-1}(B))=(g\circ f)^{-1}(B)$ , it follows that $(g\circ f)^{-1}(B)$ is open and the composition if continuous.