TITLE: The consistency of Martin's Axiom
QUESTION [10 upvotes]: In learning about the Consistency of Martin's Axiom through Kunen and Jech with help from other set theorists, I have come to a basic question about marrying these proofs:
What is the connection between the nice names argument in Kunen and the boolean valued proof in Jech regarding a limit on the number of partial order-names for a partial order which satisfies the countable chain condition?
I am more familiar with the nice names argument and trying to understand why every boolean value of an element of a set is determined by a countable anti chain in the partial order.  I can't see how values in the boolean algebra are boolean sums of the values of an anti-chain in the partial order.  Any help to understanding the link would be appreciated.

REPLY [3 votes]: I like to think about the matter topologically.
Let $X=MF(P)$ be the set of all maximal filters of $P$.
Give $X$ the topology generated by the sets
$N_p=\{U\in X: p\in U\}$. Declare $B(P)$ to be
the algebra of regular open subsets of $X$ (that is,
sets equal to the interior of their closures).
Given a regular open set $R$, pick a maximal family
$A$ of pairwise disjoint subsets of $R$ of the form
$N_p$ for some $p\in P$. The union of $A$ is dense
in $R$, so $R$ is the smallest regular open set
containing this union. In other words, every element
of $B(P)$ is the sum of an antichain from $P$.