Monotone and open Whitney maps defined on hyperspaces

Let $X$ be a nonempty continuum. It is known that there exists a map, which is called Whitney map, satisfying some special properties. Whitney map for $2^X$ need not be monotone and open. However, Whitney maps for defined on $C(X)$ i.e the closed subset of $2^X$, have these properties. One of the aims of this dissertation is to investigate whether Whitney maps defined $2^X$ are monotone and open. In addition, if the Whitney map is denoted by $\omega$, the structure of Whitney level denoted by $\omega^{-1}(t)$ (in some articles it is called Whitney contunia too) is investigated whenever X is Peano continuum. Because, locally connectedness of Whitney level for an arbitrary $t\in [0,\omega(X)]$ is not known yet. \tab This dissertation is organized as follows, we give, first, a comprehensive introduction part to explain a motivation of this dissertation after that general definitions and theorem(s) used in this thesis are given, and then the notion of a hyperspace is mentioned briefly. The existence and extension of a Whitney map is the crucial parts of this dissertation. Since our study depends on its existence, we work on it and its point inverses whenever $X$ is, especially, a locally connected continuum. Chapter six and seven are related to these properties. In the last chapter, we give some observations which is not in literature and we mention what we are working on and we also pose some questions for interested reader.