The game tree languages can be viewed as an automata-theoretic counterpart of
parity games on graphs. They witness the strictness of the index hierarchy of
alternating tree automata, as well as the fixed-point hierarchy over binary
trees. We consider a game tree language of the first non-trivial level, where
Eve can force that 0 repeats from some moment on, and its dual, where Adam can
force that 1 repeats from some moment on. Both these sets (which amount to one
up to an obvious renaming) are complete in the class of co-analytic sets. We
show that they cannot be separated by any Borel set, hence {\em a fortiori} by
any weakly definable set of trees. This settles a case left open by
L.Santocanale and A.Arnold, who have thoroughly investigated the separation
property within the μ-calculus and the automata index hierarchies. They
showed that separability fails in general for non-deterministic automata of
type Σnμ, starting from level n=3, while our result settles
the missing case n=2
