Topological Data Analysis (TDA) is a powerful mathematical theory, largely unexplored in theoretical chemistry. In this work we demonstrate how TDA provides new insights into topological features of electron densities and reduced density gradients, by investigating the effects of relativity on the bonding of the Au₄‐S‐C₆H₄‐S′‐Au′₄ molecule. Whereas recent analyses of this species carried out with the Quantum Theory of Atoms‐In‐Molecules (a previous study) concluded, from the emergence of new topological features in the electron density, that relativistic effects yielded noncovalent interactions between gold and hydrogen atoms, we show from their low persistence values (which decrease with increased basis set size) these features are not significant. Further analysis of the reduced density gradient confirms no relativity‐induced noncovalent interactions in Au₄‐S‐C₆H₄‐S′‐Au′₄. We argue TDA should be integrated into electronic structure analysis methods, and be considered as a basis for the development of new topology‐based approaches.