$x_\downarrow:=\{ y\in T\mid y<_T x\}$. Then, define $$\mathcal I:=\{ X\in[T]^{\le\aleph_0}\mid \forall x\in T(X\cap x_\downarrow)\text{ is finite }\}.$$ It is clear that $\mathcal I$ is an ideal . Let us show that moreover it is a P-Ideal, namely, that $\mathcal I$ is closed under taking pseduo-unions of countable sets. Lemma. For every