ABSTRACT
The important role played by valuation-induced topologies in the theory of topological fields was amply demonstrated in earlier chapters by theorems proved using only the most primitive properties of valuations: topologies induced by valuations are minimal (Theorem 3.3.1), locally bounded (comments following Definition 4.2.1), and completable (Corollary 6.2.2–1); a collection of proper field topologies induced by valuations is independent (Theorem 4.5.1); there is a 1–1 correspondence between valuation near orders and equivalence classes of valuations (Theorems 4.4.3 and 4.3.1(3)). In this part we develop a portion of the theory of valuations which is essential for the understanding of the theory of topological fields.
