# Thread: pathwise connected subset

1. ## pathwise connected subset

Hi,

Could someone help me prove the following:

Let $f:X \to Y$ be a continuos map between metric-spaces $(X,d_{X})$ and $(Y,d_{Y})$. Prove that if $T \subseteq X$ is a pathwise connected subset in $X$, then the image $f(T) \subseteq Y$ under $f$ is a pathwise connected subset in $Y$.

2. $f:T\rightarrow f(T)$ is surjective so, for $a,b\in f(T)$ there exist $a',b'\in T$ such that $f(a')=a$ and $f(b')=b$ . As $T$ is pathwise connected, there exists a path $\gamma$ from $a'$ to $b'$ contained in $T$. Then, $f\circ \gamma$ is a path from $a$ to $b$ contained in $f(T)$ .

Fernando Revilla

3. Originally Posted by FernandoRevilla
$f:T\rightarrow f(T)$ is surjective
By the way surjective, I suppose you understand why $f:T\rightarrow f(T)$ is surjective .