Composition of Continuous Functions is Continuous

Last updated Nov 1, 2022

Suppose $X,Y, Z$ are Topological Spaces and $F : X \to Y, G: Y \to Z$ are Continuous Functions. Then $G \circ F$ is a Continuous Function.

Let $W \subset Z$ be Open. Then because $G$ is continuous, $G^{-1}(W)$ is Open in $Y$. Since $F$ is continuous, $(G \circ F)^{-1}(W) = F^{-1}(G^{-1}(W))$ is Open in $X$. Since $W$ was arbitrary, $G \circ F$ is a Continuous Function. $\blacksquare$

• Function Preimage
• Function Composition