Abstract:
This work has the objective of presenting results related with an important class of functions (the essential functions) and theorems of separation of the euclidean space and sphere and its consequences. The proofs of the results presented demand some knowledge about polyhedron, homeomorphisms and homotopic functions. So, beyond a previous chapter, this work dedicates the chapter two to the study of polyhedron and the chapter three to results about homotopic functions. In the chapter four it is done a study about results of the separation of the euclidean space and the sphere, with relevance for the invariance of Domain Theorem and the Jordan Theorem. This chapter ends with a presentation of two examples, related with theorems presented. This examples are the Lakes of Wada, that consist in a construction of three nonempty, disjoint and open subsets of R2 that have the same boundary, and the "Alexander's Sphere" that is a subject SA of R3, homeomorphic to S2, but such that R3 \ SA and R3 \ S2 aren't homeomorphic. |