In this talk we will present a metric approach to topological data analysis that is based on a mathematical formalization of the concept of observer, seen as a collection of suitable operators acting on a metric space of functions. These functions represent the set of data that are accessible to the observer, while the operators describe the way the observer elaborates the data and enclose the invariance that he/she associates with them. In particular, we will illustrate the concept of G-equivariant non-expansive operator and how it can be used to build G-invariant persistent homology. The exposition will remain at an elementary and non-technical level, limiting itself to a description of the main ideas in our mathematical setting.