TITLE: Arc connectedness of product spaces
QUESTION [5 upvotes]: 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 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?

REPLY [7 votes]: 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.