 Vladimir Berkovich reformulated much of the theory of rigid analytic spaces in the late 1980s, using a generalization of the notion of Gelfand spectrum for commutative unital "C*"-algebras.The Berkovich spectrum of a Banach "k"-algebra "A" is the set of multiplicative semi-norms on "A" that are bounded with respect to the given norm on "k", and it has a topology induced by evaluating these semi-norms on elements of "A".Since the topology is pulled back from the real line, Berkovich spectra have many nice properties, such as compactness, path-connectedness, and metrizability.