NON-ARCHIMEDEAN ANALYTIFICATION OF ALGEBRAIC SPACES

We study quotient problems for etale equivalence relations in non-archimedean geometry, and we construct quotients for such equivalence relations in Berkovich's category of analytic spaces, assuming a separatedness hypothesis on the equivalence relation. We also give counterexamples that show the necessity of separatedness hypotheses, in contrast with the complex-analytic case. As an application, we construct analytifications for separated algebraic spaces over a non-archimedean field.