1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ). 2. Any topological space can be converted into a metric space only if there is a metric d such that the topology induced by d corresponds to an original topology. I am wondering if above statements are true or not. As per #2, if #2 is right, what topological properties with which every #2 convertible (metrizable ?) topological space share? Thanks in advance.