Can a solenoid exist in the plane?

Question

If $X$ is an $n$-dimensional continuum, then $X$ can be embedded in $\mathbb{R}^{2n+1}$.

So if $X$ is a solenoid, it can be embedded in $\mathbb{R}^3$, we even have a construction of this.

Is it possible for a solenoid to be embedded in the plane?

Answer 1

Turning my comment into an answer to remove the question from the unanswered list.

Solenoids are known to be homogeneous (in fact being homogeneous and containing only arcs as proper subcontinua characterizes solenoids among continua).

Planar homogeneous continua are classified: by a 2018 result of Hoehn-Oversteegen they are $S^1$, the pseudoarc and a space known as the circle of pseudoarcs, so solenoids are not planar (except for $S^1$ which is a very special solenoid of course).

Even without the full classification it is a 1960 result of Bing that the only homogeneous planar continuum containing an arc is $S^1$, which also rules out solenoids.