This chapter introduces the basics of differentiable manifolds and semi-Riemannian geometry for applications in general relativity, focusing on finitistic substitutes for topological concepts. It shows that many fundamental notions of semi-Riemannian geometry, such as vectors, tensors, covariant derivatives, and geodesics, are essentially finitistic. The chapter analyzes the applicability of infinite and continuous models to finite physical systems, using Hawking's singularity theorem as an example. It demonstrates that even though the theorem's classical proof is non-constructive, it can be transformed into valid logical deductions based on true premises about real spacetime, even if spacetime is discrete at the microscopic level. The conclusion of the theorem is thus physically reliable for real spacetime. The chapter follows classical presentations in Wald, O'Neill, and Naber, focusing on mathematical aspects and ignoring physical details. It introduces rational open balls and regular open subsets of $ \mathbb{R}^n $, which are defined using sequences of rational open balls. Regular open sets are used to approximate open subsets of $ \mathbb{R}^n $ in a uniform manner, as they can represent common open shapes like balls, polyhedrons, and ellipsoids. The chapter shows that regular open sets can be represented using open cubes or other regular shapes, making proofs easier. It also proves that the intersection of finitely many regular open sets is still a regular open set. The chapter concludes that regular open subsets are sufficient for representing physically meaningful open spacetime areas, as general relativity is an approximation to spacetime structure above the Planck scale. It also defines the concept of a ball being well-contained in a set and shows how regular open sets can be constructed to ensure this property.This chapter introduces the basics of differentiable manifolds and semi-Riemannian geometry for applications in general relativity, focusing on finitistic substitutes for topological concepts. It shows that many fundamental notions of semi-Riemannian geometry, such as vectors, tensors, covariant derivatives, and geodesics, are essentially finitistic. The chapter analyzes the applicability of infinite and continuous models to finite physical systems, using Hawking's singularity theorem as an example. It demonstrates that even though the theorem's classical proof is non-constructive, it can be transformed into valid logical deductions based on true premises about real spacetime, even if spacetime is discrete at the microscopic level. The conclusion of the theorem is thus physically reliable for real spacetime. The chapter follows classical presentations in Wald, O'Neill, and Naber, focusing on mathematical aspects and ignoring physical details. It introduces rational open balls and regular open subsets of $ \mathbb{R}^n $, which are defined using sequences of rational open balls. Regular open sets are used to approximate open subsets of $ \mathbb{R}^n $ in a uniform manner, as they can represent common open shapes like balls, polyhedrons, and ellipsoids. The chapter shows that regular open sets can be represented using open cubes or other regular shapes, making proofs easier. It also proves that the intersection of finitely many regular open sets is still a regular open set. The chapter concludes that regular open subsets are sufficient for representing physically meaningful open spacetime areas, as general relativity is an approximation to spacetime structure above the Planck scale. It also defines the concept of a ball being well-contained in a set and shows how regular open sets can be constructed to ensure this property.