This question (which is more a curiosity than a research problem) originates from these two:


1. <https://math.stackexchange.com/questions/720254/is-there-a-nonempty-open-bounded-subset-of-plane-whose-boundary-contains-no-1-di>
2. [complement of a totally disconnected closed set in the plane](https://mathoverflow.net/questions/55718/complement-of-a-totally-disconnected-closed-set-in-the-plane)


The first question basically asks: does there exist a non-dense, open subset of $S^2$ whose boundary contains no image of injective paths? I think that this reduces to asking for an open, non-dense set whose boundary is totally path-disconnected. 


If we remove the word "path", then the answer is given in question 2. Is the answer easy/known/unknown including the word "path"? 



Must a closed totally path-disconnected subset of the sphere have connected complement?
A circular version of the [pseudo-arc](http://en.wikipedia.org/wiki/Pseudo-arc) (where you construct it out of "circular chains" whose ends connect up to each other) is a counterexample. It is connected and totally path-disconnected, and its complement has two components. This example seems to be due to Bing (Example 2 of [this paper](http://projecteuclid.org/euclid.pjm/1102613150)).


