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Another note on countable Boolean algebras. (English) Zbl 0887.06007
For a Boolean algebra $$A$$, let $$\text{Sub}(A)$$ denote the sublattice of all subalgebras of $$A$$ endowed by the topology whose subbase is $\{\{C\in \text{Sub}(A);a\in C\},\{C\in \text{Sub} (A);a\notin C\};a\in A\}.$ It is proved that a Boolean algebra $$A$$ is countable if and only if $$\text{Sub}(A)$$ has a continuous complementation. As a consequence we obtain that a compact zero-dimensional topological space is metrizable if and only if there is a continuous mapping $$f:X\times \text{exp}(X)\rightarrow X$$ such that
$$f(x,Y)\in Y$$ for all $$x\in X$$ and all $$Y\subseteq X$$;
$$f(y,Y)=y$$ for all $$y\in Y\subseteq X$$.
##### MSC:
 600000 Structure theory of Boolean algebras 5.4e+36 Metric spaces, metrizability
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