Some Topological Properties of Certain Surfaces

Topology is one of the most important subject in mathematics, computer science and electronic physics. Perhaps the first work which deserves to be considered as the beginnings of topology is due to Euler.The title itself indicates that Euler was aware that he was dealing with a different type of geometry where distance was not relevant.Today topological ideas are present in almost all areas of today's mathematics. The book is related to topology which consists of several different branches, such as point set topology, algebraic topology and differential topology, which have relatively little in common.In the first chapter, some basic notions and results of Geometry, Convex Geometry and Elementary Geometry are introduced with some basic properties of vectors and affine space. In the second chapter, concept of curves and some local and global properties are discussed.In the third chapter, some basic notions and concept of surfaces are discussed and local properties and global properties such as geodesic are given. Finally, I have discussed some new topological properties of different surfaces and at the end discuss the Euler's Characteristics and The coverings and universal cover.