Continuous Functions Preserve Path-Connectedness

Last updated Nov 1, 2022

Let $X, Y$ be Topological Spaces and suppose $X$ is Path-Connected. Then if $f: X \to Y$ is a Continuous Function, $f(X)$ is also Path-Connected.

Let $y, y’ \in f(X)$. Then there exist $x, x’ \in X$ so that $f(x) = y$ and $f(x’) = y’$. Since $X$ is Path-Connected, there a Continuous Path $\gamma : [0,1] \to X$ so that $\gamma(0)=x$ and $\gamma(1) = x’$. Since $f$ is continuous, $f \circ \gamma$ is also a Continuous Path. Furthermore $f(\gamma(0)) = y$ and $f(\gamma(1)) = y’$, so there exists a Continuous Path from $y$ to $y’$. Since $y,y’ \in f(X)$ were arbitrary, $f(X)$ is Path-Connected. $\blacksquare$