Is a compact, connected subset of $\Bbb{R}^n$ whose boundary has empty interior inside it determined by its boundary?

Here's an example. It's a little tricky to explain, so I'll just upload a picture:

The subsets are inside $\Bbb{R}^2$. The idea is that there are infinitely many such tear-shaped 'layers', red and then blue and then red and then blue, with black in between. We take $A_1$ to be the red + black bits, and $A_2$ to be the blue + black bits. Then the boundary of both is the black, and they satisfy all the necessary conditions.

Obviously this isn't a rigorous proof, but I think writing this out in detail would be torture...