I am puzzling over the following problem, which I have been couldn't solve yet.
Suppose we are given an infinite family $\{S_i\}_{i\in I}$ of disjoint Boolean spaces - compact, Hausdorff topological spaces with base of clopen (simultaneously closed and open) subsets (alternatively called Stone spaces).
Is it always possible to find a Boolean space $S$ containing $\bigcup S_i$; and such that for each $i\in I$ subsest $S_i\subseteq S$ be a closed subset of $S$?
I have tried to consider a topology on $\bigcup S_i$ generated by $(\bigcup_{k\neq i}S_k)\cup U_j$, where $U_j\in \mathcal O(S_i)$ an open subset of $S_i$, but it is easy to see that in that case we have no $T_2$ separation. Apart that one I have no other ideas yet. Any suggestions, references and any help is appreciated very well.
Sure. For instance, you can just take $S$ to be the 1-point compactification of the disjoint union $\coprod S_i$. Or, you could take the coproduct of the $S_i$ in the category of Stone spaces, which is dual to the product $\prod B_i$ of the clopen algebras $B_i$ of each $S_i$. (This coproduct is in fact the Stone-Cech compactification of the disjoint union $\coprod S_i$, though that takes some work to prove.)
You may find it easier to think about in terms of Boolean algebras. If $B_i$ is the clopen algebra of $S_i$ and we assume for convenience that we want $\bigcup S_i$ to be dense in $S$, then we're just looking for an $S$ that is the Stone space of a subalgebra $B\subseteq \prod B_i$ with the following properties. First, the projection of $B$ onto each $B_i$ is surjective (this gives an embedding of each $S_i$ into $S$). Second, and for each distinct $i,j$, there exists $b\in B$ whose $i$th coordinate is $1$ and $j$th coordinate is $0$ (this means the images of $S_i$ and $S_j$ in $S$ are disjoint). Alternatively, you can combine these two conditions as the following condition: for each finite $F\subseteq I$, the projection $B\to\prod_F B_i$ is surjective.
The second construction of the first paragraph is given by taking $B$ to just be the full product algebra $\prod B_i$, and the first (the one-point compactification) is given by taking $B$ to be the subalgebra of $\prod B_i$ consisting of elements that are either $0$ on all but finitely many coordinates or $1$ on all but finitely many coordinates.