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Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces Representation theorems and examples Non-trivial dual notions Stone Duality and the Representation Theorem Matthew Gwynne 15th December 2008 Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Boolean Algebras and Topological Spaces Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Motivation We have seen already that fields of sets form boolean algebras. Another type of set system with similar properties to fields of sets are the topologies of topological spaces. Question Can we form boolean algebras from topological spaces? If so, how? Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Well, what do we have in an arbitrary topological space? Points in the space. Open sets and closed sets. So, given we already have the notion of a field of sets and the open sets form a nice set system, it is natural to consider such a system for the formation of a boolean algebra. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions So the topology τ of a topological space (X ; τ) is : closed under arbitrary union. closed under finite intersection. However, it is not in general closed under complementation. So we consider the largest subset of τ which is a field of sets, namely the clopen sets. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions The B functor Definition Given a topological space Y = (X ; τ), we define B(Y ) as boolean algebra formed by taking the field of clopen subsets of Y . Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Lemma B has the following properties, given the structure of all topological spaces C and the structure of all boolean algebras D : 1 For every X 2 C,B(X ) 2 D. 2 For every (f : X ! Y ) 2 C,B(f ): B(Y ) ! B(X ) 2 D 3 B(idX ) = idB(X ). 4 B(g ◦ f ) = B(f ) ◦ B(g) Translations which have such properties are called contravariant functors. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Proof. (B is a contravariant functor). (1) is trivial (it is clear that B(X ) is a field of sets). For (2), we have B(f ) = f −1, so : 1 B(f )(fg) = f −1(fg) = fg (By Def of inverse image). 2 B(f )(A0) = f −1(A0) = f −1(A)0 = B(A)0 (trivial property of total inverse maps) 3 B(f )(A [ C) = f −1(A [ C) = f −1(A) [ f −1(C) = B(f )(A) [ B(f )(C) (trivial property of inverse maps) 4 B(f )(A \ C) = f −1(A \ C) = f −1(A) \ f −1(C) = B(f )(A) \ B(f )(C) (trivial property of inverse maps) (3) is trivial : B(id ) = id−1 = id == id X X X P(X ) (4) is again trivial : B(g ◦ f ) = (g ◦ f )−1 = f −1 ◦ g −1 = B(f ) ◦ B(g). Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions So we can translate an arbitrary topological space into a boolean algebra, the question now is : Question Can we do the reverse? Can we translate a boolean algebra into a topological space? Matthew Gwynne Stone Duality and the Representation Theorem Solution Don't look into the individual boolean algebras, but consider the morphisms between them (i.e think from a category theoretical point of view). Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Problem Given such a boolean algebra, what can we actually do to form a topological space? Such structures are rather arbitrary. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Problem Given such a boolean algebra, what can we actually do to form a topological space? Such structures are rather arbitrary. Solution Don't look into the individual boolean algebras, but consider the morphisms between them (i.e think from a category theoretical point of view). Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions The Stone space functor Definition Given a boolean algebra A, the Stone space S(A) is the set of A homomorphisms from A to B2 with the induced topology from 2 . Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions As a reminder : Definition The cantor space 2A is defined as the set of all functions from A to B2 = f0; 1g with the topology such that if one fixes an a 2 A and a b 2 B2 then the set ff : f (a) = bg is an open set in the subbase of 2A, i.e all open sets of 2A may be formed from the finite intersection (to form the base) of elements in the subbase and then arbitrary unions of elements in the base. Definition Given a subset U of a topological space (X ; τ), the induced topology of U from (X ; τ) is the τU := fU \ O j O 2 τg, giving the induced topological space (U; τU ). Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Lemma S is a contravariant functor from the boolean algebras to the topological spaces. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Proof. (S is a contravariant functor) - Part 1. (1) follows from the definition. For (2), given a homomorphism f , we have S(f ): S(B) ! S(A) with S(f )(h) = h ◦ f , and must show S(f ) is continuous. It suffices to show that the inverse image of elements in the subbase of S(A) are clopen in S(B). So consider, for some a 2 A, b 2 B2 : S(f )−1(fh : h(a) = bg) = fg : S(f )(g) 2 h : h(a) = bg = fg : S(f )(g)(a) = bg = fg :(g ◦ f )(a) = bg = fg : g(f (a)) = bg So the inverse image of such a subbase element in S(A) is a subbase element in S(B). Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces From Topological Spaces to Boolean Algebras Representation theorems and examples From Boolean Algebras to Topological Spaces Non-trivial dual notions Proof. (S is a contravariant functor) - Part 2. (3) follows trivially : S(idB )(h) = h ◦ idB = h = idS(B)(h). (4) again follows trivially : S(g ◦ f ) = h ◦ g ◦ f = S(f )(h ◦ g) = S(f )(S(g)(h)) = (S(f ) ◦ S(g))(h). Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces Boolean algebras and dual algebras Representation theorems and examples Stone spaces and the Stone Duality Non-trivial dual notions Duality and Stone Spaces Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces Boolean algebras and dual algebras Representation theorems and examples Stone spaces and the Stone Duality Non-trivial dual notions The question that now arises is : Question Are these functors inverse to each other? Are the boolean algebras and topological spaces dual to each other, i.e is S(B(A)) isomorphic to A, and is S(B(X )) homeomorphic to X ? Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces Boolean algebras and dual algebras Representation theorems and examples Stone spaces and the Stone Duality Non-trivial dual notions Lemma Given a boolean algebra A, B(S(A)) is isomorphic to A. Matthew Gwynne Stone Duality and the Representation Theorem Boolean Algebras and Topological Spaces Duality and Stone Spaces Boolean algebras and dual algebras Representation theorems and examples Stone spaces and the Stone Duality Non-trivial dual notions Proof.
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