A Polish space is a topological space whose topology can be induced by a complete separable metric. The completeness requirement concerns some compatible metric; it need not hold for every metric inducing the topology. Euclidean spaces are examples.
A tree on is a subset of 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.
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A Polish space is a concept from the field of topology and descriptive set theory. Specifically, a Polish space is a topological space that is separable (contains a countable dense subset) and completely metrizable (can be endowed with a metric that induces its topology and is complete, meaning every Cauchy sequence converges within the space).