Raney Algebras and Duality for $$T_0$$ T 0 -Spaces
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Raney Algebras and Duality for T0 -Spaces G. Bezhanishvili1 · J. Harding1 Received: 14 April 2020 / Accepted: 31 July 2020 © Springer Nature B.V. 2020
Abstract In this note we adapt the treatment of topological spaces via Kuratowski closure and interior operators on powersets to the setting of T0 -spaces. A Raney lattice is a complete completely distributive lattice that is generated by its completely join prime elements. A Raney algebra is a Raney lattice with an interior operator whose fixpoints completely generate the lattice. It is shown that there is a dual adjunction between the category of topological spaces and the category of Raney algebras that restricts to a dual equivalence between T0 -spaces and Raney algebras. The underlying idea is to take the lattice of upsets of the specialization order with the restriction of the interior operator of a space as the Raney algebra associated to a topological space. Further properties of topological spaces are explored in the dual setting of Raney algebras. Spaces that are T1 correspond to Raney algebras whose underlying lattices are Boolean, and Alexandroff T0 -spaces correspond to Raney algebras whose interior operator is the identity. Algebraic description of sober spaces results in algebraic considerations that lead to a generalization of sober that lies strictly between T0 and sober. Keywords Topological space · T0 -space · T1 -space · Alexandroff space · Sober space · Closure algebra · Interior algebra Mathematics Subject Classification 54D10 · 06D22 · 06E25 · 06D10
1 Introduction Kuratowski [7] gave an alternate means to define a topology on a set X through a closure operator on the powerset, an operator ♦ that satisfies a ≤ ♦a, ♦♦a ≤ ♦a, ♦0 = 0, and ♦(a ∨b) = ♦a ∨♦b. Abstracting from such operators on powersets to Boolean algebras gave rise to the closure algebras of McKinsey and Tarski [9], who among other things showed that closure algebras are algebraic models of the modal logic S4. Because of this, closure algebras
Communicated by Jorge Picado.
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G. Bezhanishvili [email protected] J. Harding [email protected]
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New Mexico State University, Las Cruces, USA
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G. Bezhanishvili, J. Harding
are often referred to as S4-algebras in the modal logic literature. Passing to complements, one obtains an interior operator on the powerset, an operator that satisfies a ≤ a, a ≤ a, 1 = 1, and (a ∧ b) = a ∧ b. The corresponding algebras were studied by Rasiowa and Sikorski [12] under the name of topological Boolean algebras and by Blok [3] under the name of interior algebras. Tarski [13] showed that powerset Boolean algebras are characterized up to isomorphism as complete and atomic Boolean algebras. It follows from this and Kuratowski’s result mentioned above that topological spaces are characterized as interior algebras (B, ) whose underlying Boolean algebra is complete and atomic. Functions f : X → Y between sets correspond, via inverse image h = f −1 , to Boolean algebra homomorphisms h: ℘ (Y ) → ℘ (X ) between their powersets that preserve arbitra
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