Tree on a Polish space
= Tree on a Polish space
A tree on $P$ is a subset of $\bigcup_{n\geq0}P^n$ closed under initial segments. It is closed when each finite level is closed in the corresponding product topology, and well-founded when it has no infinite branch. A countable-basis approximation tree, with nested sets of shrinking diameter, reduces a closed well-founded tree on a <Polish space> to a countable well-founded tree and bounds its height by a countable ordinal.