SOME APPLICATIONS ON FINITE TOPOLOGICAL SPACES

Abstract

In this thesis, we introduce and study some properties of finite topological spaces based on the minimal neighborhood of each of its points. We consider T0 spaces. Then we give a link between finite T0 spaces and the related partial order. We give comparisons between some topological operators such as closure and interior, and some other results in general topology. We use the inverse limits as a tool for approximating certain topological spaces by finite ones. The interval [0, 1] (A Cantor set) is the completely regular modification (homeomorphic) of the inverse limit of finite spaces. We study the embedding of trees in the plane. Certain one-dimensional fractals such as self-similar sets and Julia sets of quadratic polynomials are represented as inverse limit of finite one-dimensional topological spaces. We focus on the sets with two pieces which meet in one point (tree-like fractals). We give some fractal structures generated from a numerus of computer experiments. A Locally connected quadratic Julia set J is the completely regular modification of the inverse limits of finite T0 spaces. The inverse spectrum is explicitly determined by the kneading sequences. We give a computer programme which uses to generate the approximations. The topology on approximations is determined by the neighborhoods of each point. We study the dynamics of Julia sets thought the multifunction from the approximation into itself. Finally, we give an application for some problems which appear in dynamical systems. There are some words which do not appear as prefix or initial subwords of any kneading sequence of a quadratic Julia set. The reason that is the orientation at some embeddings is not preserved. In such case, these sequences can not be realized a tree-like Julia sets in the plane.