Diversity of p-adic analytic manifolds

Abstract

The problem of the cardinality of the set of non-homeomorphic p-adic manifolds is solved. It is proved that there exist 2ℵ1 pairwise non-homeomorphic non-metrizable one-dimensional p-adic analytic manifolds of weight ℵ1. This contrasts with the single isomorphism class of metrizable manifolds of the same weight. Further, we prove that for p>2, there are 2ℵ1 pairwise non-isomorphic non-metrizable manifolds of weight ℵ1, which are homeomorphic.To demonstrate the wide variety of non-metrizable p-adic manifolds, and contrast with the situation for real analytic manifolds, we construct a range of ‘pathological’ non-metrizable p-adic manifolds.