I've seen a counter example to the converse of the Jordan curve theorem. That is given a partition of the plane consisting of a bounded open, a compact and an unbounded open set that the compact set contains a Jordan curve.
The counterexample would be the interior of Warzaw circle, the Warzaw circle itself and the exterior of it. What the Jordan curve then would need to be is the Warzaw circle itself, but that's not a Jordan curve.
Now clearly the counterexample already used the interior and exterior of the Warzaw circle and they obviously exists (although not guaranteed by the JCT).
Now the question is whether JCT could be extended so that the set of sets splitting the plane would be more than just the Jordan curves and include the Warzaw circle.
This is a corollary of the Alexander Duality Theorem that every compact subset $C\subset E^2$ such that $$ \check{H}^1(C)\ne 0 $$ separates the plane. Here $\check{H}$ denotes the Chech cohomology.
This theorem applies to both topological circles and the Warsaw circle.