Two Boolean algebras are elementarily equivalent if and only if they satisfy the same first-order statements in the language of Boolean algebras. We prove that every Boolean algebra is elementarily equivalent to the algebra of clopen subsets of a normal P-space.
We define “the category of compactifications”, which is denoted , and consider its family of coreflections, denoted . We show that is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone . A implies the assignment to each locally compact, noncompact a compactification minimum for membership in the “object-range” of . We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms...
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