Title: Fuzzy c-Means Clustering for Persistence Diagrams. Abstract: Persistence diagrams concisely represent the topology of a point cloud whilst having strong theoretical guarantees. Most current approaches to integrating topological information into machine learning implicitly map persistence diagrams to a Hilbert space, resulting in deformation of the underlying metric structure whilst also generally requiring prior knowledge about the true topology of the space. In this paper we give an algorithm for Fuzzy c-Means (FCM) clustering directly on the space of persistence diagrams, enabling unsupervised learning that automatically captures the topological structure of data, with no prior knowledge or additional processing of persistence diagrams. We prove the same convergence guarantees as traditional FCM clustering: every convergent subsequence of iterates tends to a local minimum or saddle point. We end by presenting experiments where our fuzzy topological clustering algorithm allows for unsupervised top-$k$ candidate selection in settings where (i) the properties of persistence diagrams make them the natural choice over geometric equivalents, and (ii) the probabilistic membership values let us rank candidates in settings where verifying candidate suitability is expensive: lattice structure classification in materials science and pre-trained model selection in machine learning.