NORMAL AND IRREDUCIBLE ADIC SPACES, THE OPENNESS OF FINITE MORPHISMS, AND A STEIN FACTORIZATION

Abstract We transfer several elementary geometric properties of rigid-analytic spaces to the world adic spaces, more precisely category which are locally (weakly) finite type over a non-archimedean field. This includes normality, irreducibility (in particular, irreducible components), and Stein factorization theorem. Most notably, we show that morphism in our is automatically open if target normal both source same pure dimension. Moreover, version theorem statement about connectedness fibers have not found literature or Berkovich spaces.