c.l.u.b. set and type of $T$

Question

In Shelah's book Proper and Improper forcing , what is the type (i.e. what is the underlying set of $T$ and $T_\alpha$, respectively) and relationship of $T$ and $T_\alpha$ for $\alpha\in \omega_1$ in the following snippet ? I'd know though, what is $\Gamma$, if I knew what is $T_\alpha$.

Answer 1

$T_\alpha$ is the $\alpha$-th level of the tree $T$. It’s in the little Notation section just before Theorem $\bf{6.1}$:

For an $\aleph_1$-tree $T$, $T_i$ is the $i$-th level, $T\upharpoonright i=\bigcup_{j<i}T_j$, and for $x\in T_\beta$, $\alpha\le\beta$, $x\upharpoonright\alpha$ is the unique $y\in T_\alpha$, $y\le x$.

Here $T$ is any Aronszajn tree.