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For the resistive pressure-gradient-driven turbulence model, the transition from laminar regime to fully developed turbulence is not simple and goes through several phases. For low values of the plasma parameter beta, a single quasicoherent structure forms. As beta increases, several of these structures may emerge and in turn take the dominant role. Finally, at high beta, fully developed turbulence with a broad spectrum is established. A suitable characterization of this transition can be given in terms of topological properties of the flow. Here, we analyze these properties that provide an understanding of the turbulence-induced transport and give a measure of the breaking of the homogeneity of the turbulence. To this end, an approach is developed that allows discriminating between topological properties of plasma turbulence flows that are relevant to the transport dynamics and the ones that are not. This is done using computational homology tools and leads to a faster convergence of numerical results for a fixed level of resolution than previously presented in Phys. Rev. E 78, 066402 (2008).