Is every path connected $F_\sigma$ subset of a plane an image of $[0,1)$?

Question

The title says it all. Let $A$ be a path connected $F_\sigma$ subset of a plane (or more generally $\mathbb{R}^n$). Recall that a subset is called $F_\sigma$ if it is a union of a sequence of closed sets.

Is it true that there is a continuous surjection from $[0,1)$ onto $A$? Equivalently, can $A$ be represented as a union of an increasing sequence of Peano continuums?

Note that we cannot drop "path", since $\{y=\sin(\frac{1}{x}),x>0\}\cup \{(0,0)\}$ is a connected $F_\sigma$ subset that cannot be represented as a union of an increasing sequence of continuums.

Answer 1

No, this fails even for compact subsets of $\mathbb R^2$. Namely, let $X=C\times[0,1]\cup[0,1]\times\{0\}$, where $C$ is the Cantor set. It is clearly path connected. $X$ cannot be an image of $[0,1)$, because the image of any interval $[0,a],a<1$ by this map can contain only finitely many points of $C\times\{1\}$ (because of compactness), and hence the image of $[0,1)$ can only contain countably many of them.

It might be of your interest that there is a complete topological classification of spaces which are images of $[0,1)$, namely they are the path-connected spaces which are countable unions of Hahn-Mazurkiewicz spaces (which means they are compact, Hausdorff, connected, locally connected, metrizable spaces), as shown here.