Is arc connected-ness well-behaved with respect to products?


That is -



> 
> $\prod X\_\alpha$ is arc connected iff $X\_\alpha$ is arc connected $\forall \alpha$
> 
> 
> 


In [this question](https://math.stackexchange.com/q/3779642) on MathStackexchange, an answer is provided only for the reverse implication, that is -



> 
> If $X\_\alpha$ is arc connected $\forall \alpha$, then $\prod X\_\alpha$ is arc connected
> 
> 
> 


However, I wasn't able to get an answer for the forward implication, nor have I been able to find it in any book or from Googling. So, is the forward implication true, or is there a counterexample for the same?



Arc connectedness of product spaces
This is false when the spaces are not Hausdorff. Let $X$ be the line with two origins $\{O\_1,O\_2\}$ and $Y$ be the usual line. Then $X\times Y$ is arc connected because you can pick an arc that starts at $(O\_1,y\_1)$ travels outside $\{O\_1,O\_2\}\times Y$ and then comes back to $(O\_2,y\_2)$, but $X$ itself is not arc connected.


