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McKinsey and Tarski initiated the study of interior algebras. We propose complete interior algebras as an alternative pointfree approach to topology. We term these algebras McKinsey-Tarski algebras or simply MT-algebras. Associating with each MT-algebra the lattice of its open elements defines a functor from the category of MT-algebras to the category of frames, which we study in depth. We also study the dual adjunction between the categories of MT-algebras and topological spaces, and show that MT-algebras provide a faithful generalization of topological spaces. Our main emphasis is on developing a unified approach to separation axioms in the language of MT-algebras, which generalizes separation axioms for both topological spaces and frames. [ABSTRACT FROM AUTHOR]

We introduce the concept of a Zemanian logic above S4.3 and prove that an extension of S4.3 is the logic of a Tychonoff HED-space iff it is Zemanian. [ABSTRACT FROM AUTHOR]

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]

When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α, where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α, where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that $${X \in {\bf H}_{1}}$$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that $${X \in {\bf H}_{n}}$$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$$ , and for a limit ordinal α, we have $${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. [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]

We shall define localic Krull dimension for topological spaces. In particular, a space X has the localic Krull dimension n if n is the greatest number such that X can be mapped, via a continuous and open map, onto the n-chain seen as an Alexandroff space. We shall discuss the applications of this concept in obtaining topological completeness results in modal logic. We shall also show how the localic Krull dimension is related to the Krull dimension in ring theory. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]