Dense Below

Let be a Poset and . We say that is dense below if

i.e. it is Dense at . Then is Dense iff it is dense below any .

Lemma

If is dense below and , then is dense below .

Lemma

If such that

is dense below , then is dense below .

Lemma

Let be a Forcing Partial Order. Let be a -Generic Filter over a countable Transitive Model . Let and . Then either or some is Incompatible with all elements of . Furthermore, if and is dense below then

Proof

Let

Then is Dense: let and assume so that is compatible with some . Then some has so and is dense. Thus intersects and let . If for some then . Otherwise, is Incompatible with all elements of as desired.

To see the last part, note that if and is dense below , then any has some with and thus there is some with , so and are Compatible. By the previous result, we find .