How to say a given space is a manifold? The only thing that props in my mind is to check if every open set has a euclidean coordinate chart on it. But, what if the space I am dealing with is not fully understood apriori? As in, how were the spaces of thermodynamic equilibrium states, phase and configuration spaces in classical mechanics, space-time (of general relativity) etc given a manifold structure? How to justify that every open set of these spaces correspond to an euclidean space?