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Let $A$ be the free Boolean algebra of countably many generators. As a Boolean algebra, it is isomorphic to the field $F$ of clopen subsets of its Stone space, namely, the Cantor space $X=2^\omega$.

It is known that $A$ is atomless, countably infinite, and incomplete. Since $A$ and $F$ are isomorphic, $F$ is also atomless, countably infinite, and incomplete.

Consider now the smallest subfield $F_0=\{\emptyset,X\}$ of clopen subsets of $Z$. Does $F_0$ inherit the properties of $F$, that is, is $F_0$ also atomless (trivially), countably infinite, and incomplete?

This question may be naive, but I just cannot decide whether or not $F_0$ inherits the properties of $F$, especially, if I consider that there exists a Boolean epimorphism of $A$ into $F_0$.

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  • $\begingroup$ $F$ is completely irrelevant to the question. You just have a (very simple!) Boolean algebra $F_0=\{\emptyset, X\}$. Have you tried simply answering any of your questions about this Boolean algebra directly? $\endgroup$ – Eric Wofsey Jul 31 '18 at 18:57
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No, for all three parts. $\{\emptyset,X\}$ has an atom, namely $X$. It is not countably infinite because it is finite; its cardinality is $2$. And it is complete: Any subset containing $X$ has $X$ as its least upper bound, and any subset not containing $X$ has $\emptyset$ as its least upper bound.

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