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For a generic filter $H$ in product forcing, let $G_0,G_1$ be its coordinate projections. They are filters in an ordered set. Given $p\in G_0$ and $q\in G_1$, choose separate pairs witnessing their membership and then a common strengthening in $H$. Upward closure puts $(p,q)$ in $H$, proving the displayed equality. Every ground-model dense factor set lifts to a dense product set, proving genericity of each projection over the ground model.

If $M$ is a countable transitive model, $\mathbb{P}\in M$, and $p\in \mathbb{P}$, enumerate the dense subsets of $\mathbb{P}$ belonging to $M$ and recursively choose a decreasing sequence starting below $p$ that meets each one. Its upward closure is a generic filter over $M$ containing $p$.

For a countable transitive model $M$, semantic forcing declares $p⊩\phi$ when every generic filter $G$ over $M$ containing $p$ gives $M[G]\models \phi$.

In standard notation for forcing, an order is separative if $p\nleqq q$ implies that some $r\le p$ is incompatible with $q$. Separativity does not imply an atomless forcing order: the one-point order is separative and has a forcing atom. Its only generic filter is a ground-model set. Atomlessness is the hypothesis in generic filter for an atomless order is new.