Tree on a Polish space

A tree on $P$ is a subset of ${\bigcup }_{n\ge 0}{P}^n$ 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.