Schreier groups and symmetric neighborhoods with a finite number of open components

The purpose of this investigation is to consider the group structure of Schreier groups for both general topological groups and euclidean space in particular where U is taken to have a finite number of components. Theorem 1 exibits a homomorphism from the Schreier group into the direct product of the underlying topological group and a specified finitely presented group with the components of U as generators. Theorem 2 shows that in euclidean space the given homomorphism is an isomorphism. Examples are given which illustrate the process laid out in Theorem 1.