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In this paper we prove that in various models of Martin’s Axiom there are perfectly normal, non-metrizable non-archimedean spaces of א2. A non-archimedean space is defined to be a topological space having a basis for its topology which is a tree under the inclusion relation. We will call a non-metrizable, perfectly normal non-archimedean space an archvillain. Nyikos [N] asked whether all perfectly normal non-archimedean spaces are metrizable, i.e. do archvillains exist? The only archvillains previously known were Souslin spaces or spaces including Souslin spaces. Todorcevic [T1] proved that in any model of MA+ ¬wKH , there is no archvillain of weight א1. We are going to prove that there are MA + ¬CH models in which there are archvillains essentially of bigger weight. Such models of course have no Souslin spaces. In [QT1], it is proved that every archvillain includes one of size א1 if one Mitchell collapses a supercompact cardinal to א2. There are Souslin trees in Mitchell models, however, so it is still open whether it is consistent that there are no archvillains. This work was inspired by conversations with S. Todorcevic and is part of the author’s doctoral dissertation under the supervision of Prof. F. D. Tall [Q]. We shall use combinatorial principles holding in L to create candidates for archvillains. These candidates are branch spaces of κ-Souslin trees, where κ is either א2 or inaccessible in L. Then we shall use familiar posets to specialize the initial subtrees of these κ-Souslin trees to obtain MA models in which these κSouslin trees have become archvillains of weight א2. 0. Terminology and notation Our topological and set theoretic notations are mainly standard, and can be found in [E], [K]. Definition. A κ-Souslin tree is a tree which has height κ and has no antichain of size κ. Received by the editors December 10, 1992 and, in revised form, January 29, 1998. 1991 Mathematics Subject Classification. Primary 03E05, 54E35, 54B05; Secondary 54G99, 54A35, 03E45.

Martin’s Axiom does not imply perfectly normal non-archimedean spaces are metrizable