In this review article the study of the development of relativistic cosmology and introduction of inflation in it is carried out. We study the properties of standard cosmological model developed in the framework of relativistic cosmology and the geometric structure of spacetime connected coherently with it. We examine the geometric properties of space and spacetime ingrained into the standard model of cosmology. The big bang model of the beginning of the universe is based on the standard model which succumbed to failure in explaining the flatness and the large-scale homogeneity of the universe as demonstrated by observational evidence. These cosmological problems were resolved by introducing a brief acceleratedly expanding phase in the very early universe known as inflation. Cosmic inflation by setting the initial conditions of the standard big bang model resolves these problems of the theory. We discuss how the inflationary paradigm solves these problems by proposing the fast expansion period in the early universe.