Simply Connected Compact Subsets of the Plane

If the condition of path-connectedness in the definition of simple-connectedness is relaxed, the concept of a set with no holes arises. A natural rank function is defined on the set NH of all compact sets with no holes in the plane. The rank function is obtained by associating a sequence of well-founded trees with each set in NH. It provides a natural measure of the complexity of these sets. It is also shown that the rank function is unbounded in v1 on the set SC of all compact simply connected sets in the plane. From this it will follow that NH and SC are not Borel subsets of the Polish space K R2 . of all compact sets in the plane.