These are the follow-up lecture notes of the lectures presented at the SusTrainable Summer School 2022 held at the University of Rijeka, Croatia, in July 2022. The main goal of the lectures is to provide a gentle introduction to topological data analysis because of its possible applications to sustainable software engineering. When applying topological data analysis to software system structures, one can observe additional structural metrics from the topological space which may be useful as a complement analysis technique in understanding complex software system behavior while aiming to address required quality attributes and sustainable goals. Specific challenges on addressing sustainability as software quality attribute, but also as an aspect of engineering software, are discussed. A running example is presented and how the topological data analysis can be used to analyse software structures is explained, related implementation details are discussed, and these observations are related to sustainable software and software engineering practice. The topological notions and ideas are introduced and explained on a toy example, in which the basic topological invariants are calculated explicitly, and then an exercise is provided in which the students can test their understanding and check the efficiency of their algorithms on examples of complex software structures.