
> 
> Is the boundary of an open, regular, bounded, path-connected, and simply connected set a Jordan curve?
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> 


Trying to find weakest condition on an open bounded set to apply Carathéodory's theorem. 
My bounded open sets can be assumed to be pretty well-behaved, but I wonder if the above conditions are sufficient. 



When is the boundary of an open planar set a Jordan curve?
This was addressed in a recent paper of R.L. Moore:



```
A Characterization of Jordan Regions by Properties Having no Reference to their Boundaries
Robert L. Moore
Proceedings of the National Academy of Sciences of the United States of America
Vol. 4, No. 12 (Dec. 15, 1918), pp. 364-370

```

