The aim of this paper is to establish the theoretical foundation for topological data analysis. For this purpose, we first define the concept of a simplicial complex, upon which we observe the topology inherited from the space \(\mathbb{R}^{d}\). In the second chapter, we associate vector spaces of n-chains, n-cycles and n-boundaries with the simplicial complex, along with some of the algebraic invariants of these structures, such as Euler’s characteristic and the Betti number. Finally, we define the n-th homology vector space of the simplicial complex. Furthermore, we investigate simplicial mappings, particularly simplicial approximations. We will demonstrate how a continuous mapping induces a linear operator in homology, and how a homeomorphism of topological spaces implies an isomorphism of their corresponding homological spaces. The conclusions on topology and homology will be drawn through the theorems on the reconstruction of the topology and homology of a set based on a sample. Lastly, we introduce persistent homology and persistent Betti numbers, which are relatively robust against noise in data. The paper concludes with visualizations of the evolution of topological features obtained through the Vietoris-Rips filtration.