Bijection $f: \mathbb R^n \to \mathbb R^n$ that maps connected onto connected sets must map closed connected onto closed connected sets?

Question

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 connected if 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.

Answer 1

This question is equivalent to the question of Willie Wong because of the following theorem of Jones.

Theorem (Jones, 1967). Each bijective semicontinuous map from a topological space to a semilocally locally connected Hausdorff space is continuous.

A topological space $X$ is semilocally connected if if has a base of the topology consisting of open sets whose complements have finitely many connected components.

A function $f:X\to Y$ is called

$\bullet$ Darboux if for any connected subspace $C\subset X$ the image $f(C)$ is connected in $Y$;

$\bullet$ connected if for any connected subspace $B\subset Y$ the preimage $f^{-1}(B)$ is connected in $Y$;

$\bullet$ semiconnected if for any connected closed subset $B\subset Y$ the preimage $f^{-1}(B)$ is connected and closed in $X$.

In this terms the problems of @Right and Wong read as follows:

Problem (@Right). Is each connected bijection of $\mathbb R^n$ semiconnected?

Problem (Wong). Is each connected bijection of $\mathbb R^n$ a homeomorphism?

Now I explain why these two problems are equivalent:

If the answer to the problem of @Right is affirmative, then any connected bijection $f$ of $\mathbb R^n$ is semiconnected and by Jones Theorem is continuous. By the Invariance of Domain Principle, $f$ is open and hence a homeomorphism. So, we get an affirmative answer to the problem of Wong.

If the problem of Wong has an affirmative answer, then any connected bijection $f$ of $\mathbb R^n$ is a homeomorphism and hence it is both semiconnected and Darboux.

Concerning (partial) answers to the equivalent problems of Wong and @Right let us mention the following two results:

Theorem (Tanaka, Pervine, Levine). A connected Darboux bijection $f:X\to Y$ from a Hausdorff topological space $X$ to a semilocally connected Hausdorff space $Y$ is continuous.

and

Theorem (Banakhs). A Darboux injective map $f:X\to Y$ between connected metrizable spaces is continuous if one of the following conditions is satisfied:

1) $Y$ is a 1-manifold and $X$ is compact;

2) $Y$ is a 2-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 2$;

3) $Y$ is a 3-manifold and $X$ is a closed $n$-manifold of dimension $n\ge 3$ with finite homology group $H_1(X)$.