  We define the notion of normal A-schemes, and approximable A-schemes.
Approximable A-schemes inherit many good properties of ordinary schemes. As a
consequence, we see that the Zariski-Riemann space can be regarded in two ways
-- either as the limit space of admissible blow ups, or as the universal
compactification of the given non-proper scheme. We can prove Nagata embedding
using Zariski-Riemann spaces.
