The proposed textbook belongs to the fields: probability theory and mathematical statistics, random processes, topology. The textbook contains a fairly complete variety of studied (used) spaces: topological, linear, multinormated, locally convex, countably normalized, F - spaces. Classes of functionals and random processes determined on spaces are presented, methods of metrization and convergence under additional conditions are described: linearity or convexity or randomization. The appendix contains a short and complete training (reference) course "Topological Spaces" in an accessible form for beginners. Additionally, special probabilistic designs demanded by new models are considered.