ORDER-THEORETIC INVARIANTS IN SET-THEORETIC TOPOLOGY

property (K) if every uncountable subset of P contains an uncountable linked set. We
say a subset C of P is centered if every finite subset of C has a lower bound in P. We
say P is σ-centered if it is the union of some countable family of centered sets.
Every σ-centered forcing has property (K); every forcing with property (K) is ccc. If P
is a ccc forcing and G is a P-generic over V , then V [G] preserves cardinals and cofinalities,
meaning that if α ∈ On, then the V -interpretation and V [G]-interpretation of |α| and
cf α are identical. We symbolically denote these identities by writing |α| V [G] = |α| and
(cf α) V [G] = cf α. Moreover, if A ∈ V [G] and A is an infinite subset of V , then there
is a set B ∈ V such that A ⊆ B and |A| = |B|. If P also has a dense subset of size
κ, then |λ µ | V [G] ≤ (κλ) µ for all cardinals λ and infinite cardinals µ. This is because
if P is ccc and D ⊆ P is dense, then p σ ⊆ ˇ B implies that p σ = τ for some
τ = {{ ˇ b} × Ab : b ∈ B} where each Ab is a countable antichain contained in D.
Definition 1.7.9. Martin’s Axiom, or MA, says that for every ccc forcing P and every
family D of fewer than c-many dense subsets of P, there is already in V a filter of P that
meets every dense set in D.
CH implies that D as above must be countable, so CH implies MA. Morever, Solovay
and Tennenbaum proved that if ω < κ = κ