Topological Data Analysis is an emerging field at the intersection of algebraic topology and statistical inference aimed at describing the shapes objects represented as point cloud data in the multidimensional space. Since the range of applications of shape analysis is enormous, new tests have given birth to the field of TDA. In this habilitation study three TDA-oriented tests are discussed. A new test based on metric functions is proposed. A small simulation study among the preceding tests has been employed via Monte Carlo simulation. All the mentioned tests in the vignette are activated by real world data within educational field.