We prove in this paper that there exists some infinitary rational relations
which are Sigma^0_3-complete Borel sets and some others which are
Pi^0_3-complete. This implies that there exists some infinitary rational
relations which are Delta^0_4-sets but not (Sigma^0_3U Pi^0_3)-sets. These
results give additional answers to questions of Simonnet and of Lescow and
Thomas.

This paper is a continuation of the study of topological properties of omega
context free languages (omega-CFL). We proved before that the class of
omega-CFL exhausts the hierarchy of Borel sets of finite rank, and that there
exist some omega-CFL which are analytic but non Borel sets. We prove here that
there exist some omega context free languages which are Borel sets of infinite
(but not finite) rank, giving additional answer to questions of Lescow and
Thomas [Logical Specifications of Infinite Computations, In:"A Decade of
Concurrency", Springer LNCS 803 (1994), 583-621].

We prove the following surprising result: there exist a 1-counter B\"uchi
automaton A and a 2-tape B\"uchi automaton B such that : (1) There is a model
$V_1$ of ZFC in which the omega-language $L(A)$ and the infinitary rational
relation $L(B)$ are ${\bf \Pi}_2^0$-sets, and (2) There is a model $V_2$ of ZFC
in which the omega-language $L(A)$ and the infinitary rational relation $L(B)$
are analytic but non Borel sets.

We investigate the topological complexity of non Borel recognizable tree
languages with regard to the difference hierarchy of analytic sets. We show
that, for each integer $n \geq 1$, there is a $D_{\omega^n}({\bf
\Sigma}^1_1)$-complete tree language L_n accepted by a (non deterministic)
Muller tree automaton. On the other hand, we prove that a tree language
accepted by an unambiguous B\"uchi tree automaton must be Borel. Then we
consider the game tree languages $W_{(i,k)}$, for Mostowski-Rabin indices $(i,
k)$.