The appearance of successive versions of the Gaia Data Release has prompted the realization of many works on the Pleiades, whose objectives range from the identification of new stars to the study of the cluster dynamics. Our study has a different focus, although we start by identifying a subset by imposing geometric restrictions on the m45 file from the Gaia website itself. We consider only the 2MASS stars in the file and use the stellar distances adopted in EDR3, applying restrictions concerning a high density in the spatial distribution and small distances between stars. With this denser nucleus, we carry out a study of the three-dimensional point cloud with Topological Data Analysis (TDA) techniques. We investigate its topological and geometrical structure using two TDA methods that provide different results. To clarify this problem, we introduce a dual method over a net of points where null densities are obtained and will be used to discuss loops and voids. Also, two connected components are found, and finally, we will relate the astrophysical properties of these stars for each connected component with the deduced topological-geometrical results. The small set can subsequently be expanded with stars from DR2, not 2MASS.