A theoretical overview of persistent homology

This thesis presents a theoretical overview of persistent homology, a key method in topological data analysis for capturing the evolution of homological features across a filtration. The theory of persistence modules is developed within a general framework, including the interleaving and bottleneck distances, along with their fundamental properties. The classical Nerve Theorem is proved with particular emphasis on its applicability to manifolds. The persistent homological version of this theorem provides a theoretical justification for replacing topological spaces with computationally tractable simplicial complexes over a filtration. Moreover, the stability properties of persistence modules are discussed, and several results from the literature are reviewed.