Assume that the field is complete and not discrete with respect to this valuation.
In other words, the structure sheaf of Z consists of all functions on U modulo the possible ways they can differ outside of Z.
is isomorphic (as locally ringed spaces) to an analytic variety with its structure sheaf.
The only difference is that for a scheme, the local models are spectra of rings, whereas for an analytic space, the local models are analytic varieties.
Because of this, the basic theories of analytic spaces and of schemes are very similar.
Furthermore, analytic varieties have much simpler behavior than arbitrary commutative rings (for example, analytic varieties are defined over fields and are always finite-dimensional), so analytic spaces behave very similarly to finite-type schemes over a field.
Every morphism from X to a reduced analytic space factors through r. An analytic space is normal if every stalk of the structure sheaf is a normal ring (meaning an integrally closed integral domain).
In a normal analytic space, the singular locus has codimension at least two.
When X is a local complete intersection at x, then X is normal at x. Non-normal analytic spaces can be smoothed out into normal spaces in a canonical way.
The normalization N(X) of an analytic space X comes with a canonical map ν : N(X) → X.
Every dominant morphism from a normal analytic space to X factors through ν.
Analytic spaces over algebraically closed fields are coherent.
This is not true over non-algebraically closed fields; there are examples of real analytic spaces that are not coherent.
This is often because the ground field has additional structure that is not captured by analytic sets.
But its projection onto the x-axis is the closed interval [−1, 1], which is not an analytic set.