We know that continuous functions do not preserve arc-connectedness (for an example, see this question I asked previously). So, the natural question that comes next is - which maps preserve arc-connectedness?
That is, if $X$ is arc connected, and $f:X\to Y$, then what are the weakest properties that $f$ should have so that $Y$ is arc connected. (Obviously, homeomorphisms are enough. I'm asking if there are weaker conditions that are enough.)
Every path-connected Hausdorff space is arc-connected so if you assume $X$ is arc-connected and $Y$ is Hausdorff, then $f(X)$ will be arc-connected.
The real line with "two origins" is path-connected and $T_1$ but is not arc-connected and you can construct it as the quotient of a path-connected metric space and the quotient map will even be a perfect map. This should convince you that you won't be able to do a whole lot better than this.