Building trees in large fields

We show that large rosy fields are bounded, and substantially simplify the proofs that large stable fields are separably closed and that large simple fields are bounded.

Our proofs go through in a general topological setting.

We obtain instability, non-simplicity, and non-rosiness explicitly by building the appropriate trees of definable sets.

We also show that orders on large rosy fields have several properties of orders on pseudo real closed fields.