The orderability of nonarchimedean spaces

Abstract

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F σ in X; 3. C̄ is metrizable; 4. X is orderable. It is also shown that X is orderable if C ̄ −C is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.