# $\kappa$-distributivity and Stone's Representation Theorem

On page 85 of Jech's Set Theory (3rd Edition), a complete Boolean algebra $$B$$ is defined to be $$\kappa$$-distributive if

$$$$\label{a}\tag{1} \prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i} = \sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)}$$$$

for any collection $$\{u_{\alpha, i}\}_{\alpha < \kappa, i \in I_\alpha}$$ of $$B$$. This property is easily seen to hold for a complete algebra of sets; for an arbitrary Boolean algebra, Jech presents two different characterizations of \eqref{a} in Lemma 7.16.

My question is why $$\kappa$$-distributivity is not just a consequence of Stone's Representation Theorem (Theorem 7.11): Suppose I wish to verify \eqref{a} for some Boolean algebra $$B$$. By Stone's Representation Theorem, there is an isomorphism $$\varphi$$ mapping $$B$$ to $$S$$, where $$S$$ is an algebra of sets. It suffices to verify that

$$$$\varphi\left(\prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i}\right) = \varphi\left(\sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)}\right).$$$$

To that end,

\begin{align} \varphi\left(\prod_{\alpha < \kappa}\, \sum_{i \in I_\alpha} u_{\alpha, i}\right) &= \bigcap_{\alpha < \kappa} \varphi\left(\sum_{i \in I_\alpha} u_{\alpha, i}\right)\tag{2}\\ &= \bigcap_{\alpha < \kappa}\,\bigcup_{i \in I_\alpha} \varphi\left(u_{\alpha, i}\right)\tag{3}\\ &= \bigcup_{f \in \prod_{\alpha < \kappa}} \bigcap_{\alpha < \kappa} \varphi\left(u_{\alpha, f(\alpha)}\right)\tag{4}\\ &\:\,\, \vdots\\ &= \varphi\left(\sum_{f \in \prod_{\alpha < \kappa}} \prod_{\alpha < \kappa} u_{\alpha, f(\alpha)}\right)\tag{5}, \end{align}

where (2), (3) and (5) hold because an isomorphism is a complete homomorphism; and where (4) is due to \eqref{a} and $$S$$ being an algebra of sets.

Certainly this argument is flawed, but I cannot pinpoint where I go wrong.

• Consider the algebra of clopen subsets of the Cantor set $C$. In this algebra, consider a sequence that shrinks down to one point, say $X_n=C\cap[0,3^{-n}]$. Then the intersection of these sets is $\{0\}$, but that's not clopen; it's not in the algebra under consideration. The meet in the algebra is $\varnothing$. Commented Feb 16, 2019 at 2:10