Willie Wong asked here (MO) and here (MSE) very interesting question.

As he phrased it:

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is

connectedif for every $S\subset X$ connected, $f(S)\subset Y$ is connected.

Question: Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology. Does the connectedness of (the induced power set map) $f$ imply that of $f^{-1}$?

When I did some research on that question another question became important to me, and here it is:

Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology and for every $S \subset \mathbb R^n$ connected we have that $f(S)$ is connected. Does that imply that if $T \subset \mathbb R^n$ is closed and connected that then $f(T)$ is closed and connected?

So, basically, I believe that this is really true, that is, that in requirements that $f$ maps connected sets onto connected sets and that $f$ is a bijection there is hidden a theorem that $f$ maps closed connected sets onto closed connected sets.

If this were settled we would be closer to a solution of Willie´s problem, but even if this problem stands on its own it could be of interest to someone.

secondquote box. The question in the first quote box is still an open problem $\endgroup$ – Brevan Ellefsen Aug 27 '18 at 5:39