It is classic that, to within
equivalence, the only valuations on the field F(X) of rational functions over a field F
that are improper on F are the valuations vp, where p is a prime of the
principal ideal subdomain F[X] of F(X), and the valuation v∞, defined by the
prime X−1 of the principal ideal subdomain F[X−1] of F(X) ([1], p. 94,
Corollary 2). If 𝒯 is the supremum of finitely many of the associated valuation
topologies, then 𝒯 is a Hausdorff, locally bounded ring topology on F(X)
for which F is a bounded set and for which there is a nonzero topological
nilpotent a. In this paper we shall show conversely that any topology on
F(X) having these properties is the supremum of finitely many valuation
topologies.
