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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 \(y\in Y\subseteq X\).
06E05 Structure theory of Boolean algebras
54E35 Metric spaces, metrizability
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