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

For the product forcing order, let

G_{0}

and

G_{1}

H

G_{0} = {p : \exists q ((p, q) \in H)}, G_{1} = {q : \exists p ((p, q) \in 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^{'}) \in H

and

(p^{'}, q) \in H

. Directedness supplies

(r, s) \in H

below both, so

(r, s) \leq (p, q)

. Upward closure of

H

gives

(p, q) \in H

H = G_{0} \times G_{1} .

No greatest conditions in the factors are required for this argument.

 Ancestors (11)

There are no discussions about this article yet.

There are currently no matching articles.