Simply connected sets

We shall attempt to construct a topological definition of simply connected which is applicable to any connected set. The formulation is made in terms of separation properties intrinsic to the set. In order that a connected and locally arcwise connected subset M of the plane be simply connected under the proposed definition, it is necessary and sufficient that the interior of every simple closed curve lying in M be a subset of M. The notion of simple connectedness in the weak sense is also defined. This property is possessed by every compact plane continuum which does not separate the plane, and by certain continua which do separate the plane. The properties of simply connected sets seem to have a variety of applications, and to furnish an interesting background for a number of well known theoremsMathematic