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We generalize the concept of a strong inclusion on a biframe [Sch93] to that of a proximity on a biframe, which is related to the concept of a strong bi-inclusion on a frame introduced in [PP12b]. We also generalize the concept of a bi-compactification of a biframe [Sch93] to that of a compactification of a biframe, and prove that the poset of compactifications of a biframe L is isomorphic to the poset of proximities on L. As a corollary, we obtain Schauerte's characterization of bi-compactifications of a biframe [Sch93]. In the spatial case this yields Blatter and Seever's characterization of compactifications of completely regular ordered spaces [BS76] and a characterization of bi-compactifications of completely regular bispaces. [ABSTRACT FROM AUTHOR]

We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space X and a totally ordered algebra A , we introduce the concept of a finitely valued normal function f : X → A . We show that the operations of A lift to the set FN ( X , A ) of all finitely valued normal functions, and that there is a canonical proximity relation ≺ on FN ( X , A ) . This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain A are axiomatized as proximity Baer Specker A -algebras; that is, the pairs ( S , ≺ ) , where S is a torsion-free A -algebra generated by its idempotents which is a Baer ring, and ≺ is a proximity relation on S . We introduce the category of proximity Baer Specker A -algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker A -algebras. [ABSTRACT FROM AUTHOR]

Abstract: We introduce zero-dimensional proximities and show that the poset of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that is isomorphic to the poset of Boolean bases of X, and derive Dwinger''s theorem that is isomorphic to as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone–Čech compactification of X is a unique up to equivalence extremally disconnected compactification of X. [Copyright &y& Elsevier]

We show that a topological space is hereditarily irresolvable if and only if it is Hausdorff-reducible. We construct a compact irreducible T1-space and a connected Hausdorff space, each of which is strongly irresolvable. Furthermore, we show that the three notions of scattered, Hausdorff-reducible, and hereditarily irresolvable coincide for a large class of spaces, including metric, locally compact Hausdorff, and spectral spaces. [Copyright &y& Elsevier]