Question

Prove that composition of continuous functions from topological spaces to topological spaces is continuous

Answer 1

Let $f:X\to Y$ and $g:Y\to Z$ be a continuous functions of topological spaces. Consider their composition $g\circ f: X\to Z.$ Let $U\subset Z$ be an arbitrary open set. Since $g$ is continuous, the preimage $V=g^{-1}(U)$ is open set in $Y.$ Then using the continuity of $f$, we conclude that $f^{-1}(V)$ is an open set in $X.$ Taking into accont that $(g\circ f)^{-1}(U)=f^{-1}(g^{-1}(U))=f^{-1}(V),$ we conclude that the preimage of any open set $U\subset Z$ is open set in $X,$ and hence the composition $g\circ f: X\to Z$ is continuous.