Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 121 / 3 / iv / a / Solution

For the product forcing order, let $G_0$ and $G_1$ be the coordinate projections of $H$:These are nonempty upward-closed, directed filters in an ordered set. Clearly $H\subseteq G_0\times G_1$. If $p\in G_0$ and $q\in G_1$, choose $(p,q^{\prime })\in H$ and $(p^{\prime },q)\in H$. Directedness supplies $(r,s)\in H$ below both, so $(r,s)\le (p,q)$. Upward closure of $H$ gives $(p,q)\in H$. Therefore the projection of a product-generic filter satisfiesNo greatest conditions in the factors are required for this argument.