Classically, the Riemann
mapping theorem states that any open, simply connected and proper subset of U of
the complex plane is analytically equivalent to the open unit disk S(0,1). However
this theorem is not constructively valid without some additional restriction on U.
Two separate geometric conditions, mappability and maximal extensibility, on U are
then proposed. The two conditions are shown to be mathematically equivalent.
Finally the mappability condition is shown to be both necessary and sufficient for an
analytic equivalence to exist constructively between U and S(0,1). The mappability
condition is due Errett Bishop. The sufficiency proof is based on methods contained
in [1].
