Size functions

Abstract

Abstract.We introduce the notion of a nonarchimedean size function similar to the notion of a size function introduced by Marcos. We describe a class of ring topologies on fields that are complete, neither first countable nor locally bounded, but have topologically nilpotent elements.Two types of valuations are commonly considered: real-valued valuations (also called absolute values) and nonarchimedean valuations (also called Krull valuations). In [3] Marcos introduced axioms for a function which we will call a real-valued size function. We introduce the notion of a nonarchimedean size function. As for valuations, the classes of real and nonarchimedean size functions overlap, but neither subsumes the other.Marcos introduced size functions to construct topologies which yield an affirmative answer to a thirty year old open question in [2]: Do there exist topological fields which are not locally bounded but have topologically nilpotent elements? We describe here another class of ring topologies on fields with these properties.A nonzero element x in a topological ring is called topologically nilpotent if x n → 0. A * denotes the set of nonzero elements of a subset A of an additive group, and G ≥a (resp. G >a ) denotes the set of elements greater than or equal to (resp. strictly greater than) a in an ordered group G. In an ordered group (in particular, the real numbers) a ∨ b (resp. a ∧ b) denotes the larger (resp. smaller) of a and b.