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2 Least upper and greatest lower bounds
3 Boolean algebras
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arbitrary neighborhood Axiom Boolean ring Choose closed sets closed subset closure complete Boolean algebra completely regular completely regular space contains continuous function continuous mapping converges countably additive defined denote the set dense element q G A exists an element finite subset follows fundamental sequence Hausdorff Hausdorff space Hence Hewitt space holds homomorphism of algebras implies inclusion indecomposable inequality isomorphic least upper bound metric space natural number natural topology neighborhood of zero nonvoid intersection obtain obvious open set pairwise disjoint partially ordered positive real number product topology Proposition prove pseudometrics relation relative topology satisfying the condition sequence of type set Q simplex solid neighborhood strictly positive elements subsemifield subspace Suppose theorem Tihonov semifield topological Boolean algebra topological ring topological semifield topological space ultracomplete uniform structure write