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Let E(2) be two-dimensional Euclidean space with its standard metric topology. If J is a Jordan curve which is a subset of E(2), then the bounded component of the complement of J (in E(2)) is simply connected and uniformly locally connected. Is the converse true? If K is a non-empty bounded open subset of E(2) that is also (1)connected (2)simply connected and (3)uniformly locally connected, is the boundary of K (in E(2)) always a Jordan curve?................The definition of "Uniformly locally connected-(ulc)" is as follows. A non-empty open subset S of E(2) is ulc if and only if, given any e>0 there exists d>0 such that each pair of points of S whose distance apart is less than d, belong to a connected subset of S whose diameter is less than e.

Garabed Gulbenkian
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    You should add the definition of (3) since most people do not know it. – Moishe Kohan Oct 19 '16 at 21:34
  • This probably follows from Caratheodory's theorem. I would try to show that $\partial K$ is locally connected. Related: https://math.stackexchange.com/questions/1781090/converse-to-the-jordan-curve-theorem?noredirect=1&lq=1 – Moishe Kohan Oct 19 '16 at 21:37
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    Definition of "Uniformly locally connected-(ulc)". A non-empty open subset S of E(2) is ulc if and only if, given any e>0 there exists a d>0, such that each pair of points of S whose distance apart is less than d belong to a connected subset of S whose diameter is less than e. – Garabed Gulbenkian Oct 22 '16 at 15:17
  • Right. Consider adding the definition to the main body of the question. – Moishe Kohan Oct 22 '16 at 17:05
  • @Moishe Cohen: Many thanks for the reference. – Garabed Gulbenkian Oct 22 '16 at 19:34

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