Proposition 3

The image by a continuous mapping of an arcwise connected space is arcwise connected.

Let X be a path-connected space, $f: X\rightarrow Y$ a continuous mapping. Let $x$ and $y$ be two points of $f(X)$; let $x'$ and $y'$ be points of X such that $f(x') =x$ and $f(y') =y$. There exists a path $c$ in X whose origin is $x'$ and whose term is $y'$. Then, $f\circ c$ is a path in $f(X)$ with origin $x$ and term $y$.