Full paper in PDF:
$%K. Boussaf, N. Maïnetti, and M. Hemdaoui, Tree structure on the set of multiplicative semi-norms
of Krasner algebras H(D), Rev. Mat. Complut. 13 (2000), no. 1, 85–109.%$
Laboratoire de Mathématiques Pures Université Blaise Pascal (Clermont-Ferrand) Complexe Scientifique des Cézeaux F 63177 Aubiere Cedex — France | Département de Mathématiques Pures Université Mohammed I Oujda — Morocco |
ABSTRACT
Let be an algebraically closed field, complete for an ultrametric absolute value, let be an infinite subset of and let be the set of analytic elements on (Escassut, 1995). We denote by the set of semi-norms y of the -vector space which are continuous with respect to the topology of uniform convergence on and which satisfy further whenever and . This set is provided with the topology of simple convergence. By the way of a metric topology thinner than the simple convergence, we establish the equivalence between the connectedness of , the arc-connectedness of and the infraconnectedness of . This generalizes a result of Berkovich given on affinoid algebras (Berkovich, 1990). Next, we study the filter of neighborhoods of an element of , and we give a condition on the field K such that this filter admits a countable basis. We also prove the local arc-connectedness of when is infraconnected. Finally, we study the metrizability of the topology of simple convergence on and we give some conditions to have an equivalence with the metric topology defined above. The fundamental tool in this survey consists of circular filters.
1991 Mathematics Subject Classification: 46S10, 11Q25.