On Boolean algebras and their role in analysis

The relations between complete and $\sigma$-complete covers of a Boolean algebra are examined. The Dedekind completion of a Boolean algebra is shown to be a quotient of any complete cover. Atoms of a Boolean algebra correspond to atoms of the Dedekind completion hence the Dedekind completion of an atomic Boolean algebra is isomorphic to the power set of the set of all atoms. There exists a correspondence between complete (sigma-complete) homomorphisms and full (sigma-complete) ideals. The explicit form of the Dedekind completion is given for the Boolean algebra generated by all semiopen subintervals of [0,1) as the atomless, complete Boolean algebra of all regularly closed subsets of [0,1). A compatible topology for a Boolean algebra is a topology for which addition and multiplication are continuous. The properties concerning products, quotients, subspaces and uniform completions of topological Boolean algebras are examined. Compact algebras are isomorphic and homeomorphic with power sets, endowed with the product topology. Measure algebras endowed with the weak* topology are compatible if and only if the underlying measure is purely atomic. A new proof of Stone Representation Theorem for a field of sets is given, providing a tool for establishing relations between Stone representation spaces of algebras, covers, subalgebras and quotients.