How does one prove that the boundary of a bounded open connected subset of $\mathbb{R}^2$ must contain the image of a path?
I require a non-constant path. But it doesn't have to be injective.
This thought occurred to me just now. I've looked for an answer on Google and found none.
$\\$
Would I be right in thinking that the proof might involve partitioning the plane into small squares of side length $\dfrac{1}{2^n}$, then filling in every square that overlaps with that open set, then tracing the "outer edge" of this approximation of the open set (easier to define on a shape made up of squares, I suppose), then proving that this passes to a limit, then proving that this limit is on the boundary of the open set?
I don't mind looking up other results, if the result that I want would follow from them. It's been a while since I last read a proof of the Jordan Curve Theorem. I'm asking my question here, in case there is a simple answer I've missed.
Cheers!