This chapter focuses on the connected sum of oriented surfaces. It deals with a theorem about the Euler characteristic of a connected, closed, compact two-dimensional surface F. The chapter discusses two-dimensional surfaces, the Euler characteristic, and genus. Compact, closed, two-dimensional, oriented surfaces are classified by calculating the genus. The term annulus refers to the twice punctured sphere. To form the annulus, we remove two disks from the surface of the sphere. The chapter identifies the boundaries of two-dimensional surfaces with boundary to create new surfaces. The projective plane is a surface formed by gluing the boundary of a disk to the boundary of the Mobius band. To identify a two dimensional surface, our first goal is to divide the surface into standardized components. These components are: points, line segments, and disks. Construct a sequence of surface link pairs that prove that virtual crossing number is not bounded by the virtual genus.