composition of continuous mappings is continuous

Theorem 1.

The composition of two continuous mappings (whendefined) is continuous.

Proof.

Let $X,Y,Z$ be topological space, and let $f,g$ bemappings

$f:X$$\to$$Y,$

$g:Y$$\to$$Z.$

We wish to prove that $g\circ f$ is continuous. Suppose $B$ isan open set in $Z$.Since $g$ is continuous, $g^{-1}(B)$ is an open set in $Y$,andsince $f$ 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 followsthat ${(g\circ f)}^{-1}(B)$ is open and the composition if continuous.∎