Topological Characters of Complete Metric Spaces

In this thesis the topological aspects of complete metric spaces are studied. Complete metric spaces, Characterizations of complete metric spaces and examples of complete metric spaces are presented. A completion of a metric space is discussed. Function spaces and their topologies are defined and studied. Completeness of function spaces is considered. As an application, a construction of the well-known Peano space –filling curve is discussed. Finally, Theorems of topological characters concerning complete spaces such as Heine-Borel theorem, Ascoli's theorem and Baire's theorem with their proofs are introduced, in addition to other matters concerning the subject. The existence of continuous nowhere-differentiable real-valued functions is proved.