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Topological ordered spaces as a foundation for a quantum spacetime theory Ettore Minguzzi Università Degli Studi Di Firenze Castiglioncello, September 19, 2012 Talk based on • Normally preordered spaces and utilities, Order, to appear. • Topological conditions for the representation of preorders by continuous utilities, Appl. Gen. Topol. 13, 81-89 (2012). • Quasi-pseudo-metrization of topological preordered spaces, Topol. Appl. 159, 2888-2898 (2012). Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 1/22 Ideas for a quantum spacetime theory Physical theories rely on a hierarchy of specialized mathematical structures • (pseudo)-Riemannian structure (causal order) • Differential structure (smoothness) • Topological structure (closeness and continuity) Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 2/22 Ideas for a quantum spacetime theory Physical theories rely on a hierarchy of specialized mathematical structures • (pseudo)-Riemannian structure (causal order) • Differential structure (smoothness) • Topological structure (closeness and continuity) History of science suggests: let them be dynamical Some of these levels have become dynamical in the evolution of science: e.g. the rigid Euclidean space metric of classical mechanics has been replaced by the the dynamical Lorentzian metric of general relativity. We expect that the same process should continue. Thus we must start removing structure from our theory trying to preserve physical content. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 2/22 Ideas for a quantum spacetime theory Physical theories rely on a hierarchy of specialized mathematical structures • (pseudo)-Riemannian structure (causal order) • Differential structure (smoothness) • Topological structure (closeness and continuity) History of science suggests: let them be dynamical Some of these levels have become dynamical in the evolution of science: e.g. the rigid Euclidean space metric of classical mechanics has been replaced by the the dynamical Lorentzian metric of general relativity. We expect that the same process should continue. Thus we must start removing structure from our theory trying to preserve physical content. Quantum regime Differentiability does not hold at the fundamental level, we expect it to be an emerging phenomenon. Thus we are left with • Topology and Order Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 2/22 Topology, Order and their unification Main ingredients • Topology: gives as a notion of continuity to work with. • Order: Will be interpreted in a causal fashion, thus causality is still meaningfull at a quantum regime. • (Measure): The σ-algebra will follow from the topology. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 3/22 Topology, Order and their unification Main ingredients • Topology: gives as a notion of continuity to work with. • Order: Will be interpreted in a causal fashion, thus causality is still meaningfull at a quantum regime. • (Measure): The σ-algebra will follow from the topology. Difference with Causal Set Theory We do not infer topology from order, rather we impose compatibility conditions between topology and order. Causal set theory is less general because from the order you can define only few topologies: Alexandrov, Scott, etc. Any causal set is a topological ordered space, but not conversely. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 3/22 Topology, Order and their unification Main ingredients • Topology: gives as a notion of continuity to work with. • Order: Will be interpreted in a causal fashion, thus causality is still meaningfull at a quantum regime. • (Measure): The σ-algebra will follow from the topology. Difference with Causal Set Theory We do not infer topology from order, rather we impose compatibility conditions between topology and order. Causal set theory is less general because from the order you can define only few topologies: Alexandrov, Scott, etc. Any causal set is a topological ordered space, but not conversely. Role of quasi-uniformities Topology and Order are two faces of this mathematical entity. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 3/22 Classical spacetime physics motivation: Relationship between time and closed relations Stable Causality (M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal. Here g 0 > g if the light cones of g are everywhere strictly larger than those of g. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 4/22 Classical spacetime physics motivation: Relationship between time and closed relations Stable Causality (M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal. Here g 0 > g if the light cones of g are everywhere strictly larger than those of g. The causal relation J + = {(x, y) : there is a causal curve connecting x and y} is not both closed and transitive Seifert’s relation JS+ = T g 0 >g Jg+ (1971) JS+ is closed, transitive and contains J + . Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 4/22 Classical spacetime physics motivation: Relationship between time and closed relations Stable Causality (M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal. Here g 0 > g if the light cones of g are everywhere strictly larger than those of g. The causal relation J + = {(x, y) : there is a causal curve connecting x and y} is not both closed and transitive Seifert’s relation JS+ = T g 0 >g Jg+ (1971) JS+ is closed, transitive and contains J + . Theorem The spacetime is stably causal if and only if JS+ is antisymmetric. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 4/22 Classical spacetime physics motivation: Relationship between time and closed relations Stable Causality (M, g) is stably causal if there is g 0 > g with (M, g 0 ) causal. Here g 0 > g if the light cones of g are everywhere strictly larger than those of g. The causal relation J + = {(x, y) : there is a causal curve connecting x and y} is not both closed and transitive Seifert’s relation JS+ = T g 0 >g Jg+ (1971) JS+ is closed, transitive and contains J + . Theorem The spacetime is stably causal if and only if JS+ is antisymmetric. Theorem In a stably causal spacetime the Seifert relation coincides with the smallest closed and transitive relation containing J + (also denoted K + by Sorkin and Woolgar). Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 4/22 Powerfulness of topological ordered space theory Time function A continuous real function such that p < q ⇒ t(p) < t(q). Hawking proved: Stable causality ⇒ time function. Lorentzian geometry is actually not needed in the previous problem on the existence of time functions. We shall see that Levin’s theorem Every second countable locally compact space endowed with a closed order (e.g. a stably causal spacetime endowed with the relation JS+ ) admits a strictly increasing continuous function. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 5/22 Powerfulness of topological ordered space theory Time function A continuous real function such that p < q ⇒ t(p) < t(q). Hawking proved: Stable causality ⇒ time function. Lorentzian geometry is actually not needed in the previous problem on the existence of time functions. We shall see that Levin’s theorem Every second countable locally compact space endowed with a closed order (e.g. a stably causal spacetime endowed with the relation JS+ ) admits a strictly increasing continuous function. Thus we can infer an important theorem on the existence of time functions without using the differentiable structure. It seems that we are going in the right direction. . . Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 5/22 Preorders Relation A relation on a set E is a subset R ⊂ E × E. A special relation is the diagonal ∆ = {(x, x) : x ∈ E}. It is an identity for the composition ◦ of relations. We define: R−1 = {(x, y) ∈ E : (y, x) ∈ R}. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 6/22 Preorders Relation A relation on a set E is a subset R ⊂ E × E. A special relation is the diagonal ∆ = {(x, x) : x ∈ E}. It is an identity for the composition ◦ of relations. We define: R−1 = {(x, y) ∈ E : (y, x) ∈ R}. Preorder and order • A preorder on a set E is a reflexive and transitive relation on E. • A preorder which is antisymmetric “x ≤ y and y ≤ x ⇒ x = y” is an (partial) order (R ∩ R−1 = ∆). We shall write x ≤ y for (x, y) ∈ R. Then R is called the graph of ≤. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 6/22 Preorders II Increasing/decreasing hulls Let S ⊂ E, the increasing and decreasing hulls are i(S) = {y ∈ E : x ≤ y for some x ∈ S}, d(S) = {y ∈ E : y ≤ x for some x ∈ S}. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 7/22 Preorders II Increasing/decreasing hulls Let S ⊂ E, the increasing and decreasing hulls are i(S) = {y ∈ E : x ≤ y for some x ∈ S}, d(S) = {y ∈ E : y ≤ x for some x ∈ S}. Increasing/decreasing, monotone sets Subsets S for which i(S) = S are called increasing, while subsets for which d(S) = S are called decreasing. Increasing and decreasing sets are monotone. The complement of an increasing set is decreasing and the other way around. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 7/22 Preorders III Convex set A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and only if it is the intersection of an increasing set B and a decreasing set A. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 8/22 Preorders III Convex set A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and only if it is the intersection of an increasing set B and a decreasing set A. Discrete order • If R = ∆ the order is discrete. This means i(x) = d(x) = {x}. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 8/22 Preorders III Convex set A set S is convex if S = i(S) ∩ d(S). As a consequence a set S is convex if and only if it is the intersection of an increasing set B and a decreasing set A. Discrete order • If R = ∆ the order is discrete. This means i(x) = d(x) = {x}. Isotone functions A function between two preordered spaces f : E → E 0 is isotone if x ≤ y ⇒ f (x) ≤ f (y) and anti-isotone if x ≤ y ⇒ f (y) ≤ f (x). Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 8/22 Topological preordered space Topological preordered space A topological preordered space is triple (E, T , ≤) given by a topological space (E, T ) endowed with a preorder ≤. If R = ∆ we want to recover the usual topology. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 9/22 Topological preordered space Topological preordered space A topological preordered space is triple (E, T , ≤) given by a topological space (E, T ) endowed with a preorder ≤. If R = ∆ we want to recover the usual topology. Upper and lower topologies • U = T ] is the topology generated by the open increasing sets, • L = T [ is the topology generated by the open decreasing sets. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 9/22 Topological preordered space Topological preordered space A topological preordered space is triple (E, T , ≤) given by a topological space (E, T ) endowed with a preorder ≤. If R = ∆ we want to recover the usual topology. Upper and lower topologies • U = T ] is the topology generated by the open increasing sets, • L = T [ is the topology generated by the open decreasing sets. Bitolopogy The theory of topological preordered spaces is very much connected with the theory of bitopological spaces. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 9/22 Compatibility conditions between topology and preorder I Local convexity The topology T admits a base of convex neighborhoods. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 10/22 Compatibility conditions between topology and preorder I Local convexity The topology T admits a base of convex neighborhoods. Not all open and convex sets are the intersection of and increasing open set and a decreasing open set. Convexity The topology T is generated by U and L : T = sup(U , L ). Convexity is much more important than local convexity! Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 10/22 Compatibility conditions between topology and preorder II 1 Quasi-pseudo-metrizable preordered space 2 Normally preordered space 3 Completely regularly preordered space (Tychonoff-preordered space) (Quasi-uniformizable space) 4 Closed preordered space There are some gaps in the expected relative strengths, for instance 1 does not imply 2, and convexity is needed to infer that 2 implies 3. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 11/22 Closed preordered spaces Closed preordered space E is a closed preordered space if the graph of the preorder ≤ is closed in the product topology of E × E. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 12/22 Closed preordered spaces Closed preordered space E is a closed preordered space if the graph of the preorder ≤ is closed in the product topology of E × E. The Hausdorff condition is equivalent to the statement: “∆ is closed in the product topology”. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 12/22 Closed preordered spaces Closed preordered space E is a closed preordered space if the graph of the preorder ≤ is closed in the product topology of E × E. The Hausdorff condition is equivalent to the statement: “∆ is closed in the product topology”. Proposition • Every closed preordered space in which R = ∆ is Hausdorff (T2 ). • Every closed ordered space is Hausdorff. (because R ∩ R−1 = ∆) Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 13/22 Closed preordered spaces Closed preordered space E is a closed preordered space if the graph of the preorder ≤ is closed in the product topology of E × E. The Hausdorff condition is equivalent to the statement: “∆ is closed in the product topology”. Proposition • Every closed preordered space in which R = ∆ is Hausdorff (T2 ). • Every closed ordered space is Hausdorff. (because R ∩ R−1 = ∆) General philosophy Do not assume Hausdorffness and any other usual separability conditions! We avoid to double the preorder-separability conditions. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 13/22 Normally preordered spaces Normally preordered space E is a normally preordered space if it is a closed preordered space and for every pair of closed increasing set B, and closed decreasing set A such that A ∩ B = ∅, there are an open increasing set V and an open decreasing set U such that, A ⊂ U , B ⊂ V , U ∩ V = ∅. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 14/22 Normally preordered spaces Normally preordered space E is a normally preordered space if it is a closed preordered space and for every pair of closed increasing set B, and closed decreasing set A such that A ∩ B = ∅, there are an open increasing set V and an open decreasing set U such that, A ⊂ U , B ⊂ V , U ∩ V = ∅. Theorem (Nachbin’s extension of Urysohn’s lemma) The topological preordered space (E, T , ≤) is normally preordered if and only if for any two disjoint closed subsets A, B ⊂ E, A is decreasing and B is increasing, there exist on E a continuous isotone real-valued function f such that f (x) = 0 (x ∈ A), f (x) = 1 (x ∈ B), and 0 ≤ f (x) ≤ 1 (x ∈ E). Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 15/22 Completely regularly preordered spaces Completely regularly preordered space (Tychonoff-preordered space) E is a completely regularly preordered space if (i) The topology is the weak topology of the family of continuous isotone functions. (ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y) Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 16/22 Completely regularly preordered spaces Completely regularly preordered space (Tychonoff-preordered space) E is a completely regularly preordered space if (i) The topology is the weak topology of the family of continuous isotone functions. (ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y) Theorem Every completely regularly preordered space is convex Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 16/22 Completely regularly preordered spaces Completely regularly preordered space (Tychonoff-preordered space) E is a completely regularly preordered space if (i) The topology is the weak topology of the family of continuous isotone functions. (ii) x ≤ y ⇔ ∀f continuous and isotone, f (x) ≤ f (y) Theorem Every completely regularly preordered space is convex Theorem Every completely regularly preordered space can be compactified. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 16/22 How do we improve separability properties? In usual (discrete order) topology the following results are essential Theorem Every locally compact Hausdorff space is completely regular (Tychonoff). This is usually proved using the Alexandrov one-point compactification. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 17/22 How do we improve separability properties? In usual (discrete order) topology the following results are essential Theorem Every locally compact Hausdorff space is completely regular (Tychonoff). This is usually proved using the Alexandrov one-point compactification. Using the fact that completely regular implies regular, we make the next jump with Urysohn’s metrization theorem Every second countable regular space is metrizable. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 17/22 How do we improve separability properties? In usual (discrete order) topology the following results are essential Theorem Every locally compact Hausdorff space is completely regular (Tychonoff). This is usually proved using the Alexandrov one-point compactification. Using the fact that completely regular implies regular, we make the next jump with Urysohn’s metrization theorem Every second countable regular space is metrizable. None of these results holds in the preordered case. I used a different path proving Theorem Every locally compact σ-compact space equipped with a closed preorder is normally preordered. It is an essential generalization for the manifold case. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 17/22 Quasi-pseudo-metrizability Quasi-pseudo-metric A quasi-pseudo-metric on a set X is a function p : X × X → [0, +∞) such that for x, y, z ∈ X (i) p(x, x) = 0, (ii) p(x, z) ≤ p(x, y) + p(y, z). The function q(x, y) := p(y, x) is the conjugate quasi-pseudo-metric. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 18/22 Quasi-pseudo-metrizability Quasi-pseudo-metric A quasi-pseudo-metric on a set X is a function p : X × X → [0, +∞) such that for x, y, z ∈ X (i) p(x, x) = 0, (ii) p(x, z) ≤ p(x, y) + p(y, z). The function q(x, y) := p(y, x) is the conjugate quasi-pseudo-metric. Quasi-pseudo-metrizable space A topological preordered space (E, T , ≤) is quasi-pseudo-metrizable if there is a pair of conjugate quasi-pseudo-metrics p, q, said admissible, such that T is the topology generated by the pseudo-metric p + q, and the graph of the preorder is given by G(≤) = {(x, y) : p(x, y) = 0}. Every quasi-pseudo-metrizable preordered space is a completely regularly preordered space. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 18/22 Quasi-pseudo-metrizability and ordered Hilbert cube Theorem The following conditions are equivalent for a topological preordered space (E, T , ≤) (a) (E, T , ≤) is a second countable completely regularly preordered space, (b) (E, T , ≤) is separable and quasi-pseudo-metrizable. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 19/22 Quasi-pseudo-metrizability and ordered Hilbert cube Theorem The following conditions are equivalent for a topological preordered space (E, T , ≤) (a) (E, T , ≤) is a second countable completely regularly preordered space, (b) (E, T , ≤) is separable and quasi-pseudo-metrizable. Ordered Hilbert cube The Hilbert cube H = [0, 1]N once endowed with the product topology and the product order is a strict quasi-pseudo-metric ordered I-space with P n quasi-pseudo-metric p(x, y) = ∞ n=1 max(xn − yn , 0)/2 . Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 19/22 Quasi-pseudo-metrizability and ordered Hilbert cube Theorem The following conditions are equivalent for a topological preordered space (E, T , ≤) (a) (E, T , ≤) is a second countable completely regularly preordered space, (b) (E, T , ≤) is separable and quasi-pseudo-metrizable. Ordered Hilbert cube The Hilbert cube H = [0, 1]N once endowed with the product topology and the product order is a strict quasi-pseudo-metric ordered I-space with P n quasi-pseudo-metric p(x, y) = ∞ n=1 max(xn − yn , 0)/2 . Order embedding into the ordered Hilbert cube The following conditions are equivalent for a topological ordered space (E, T , ≤) (a) (E, T , ≤) is a second countable completely regularly ordered space, (b) (E, T , ≤) is order embeddable in the ordered Hilbert cube H. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 19/22 Convexity? From local convexity to convexity A locally convex closed preordered space which is topologically a locally compact σ-compact space is necessarily convex. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 20/22 Convexity? From local convexity to convexity A locally convex closed preordered space which is topologically a locally compact σ-compact space is necessarily convex. Justifying local convexity Every compactly generated closed ordered space is locally convex. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 20/22 Conclusions • We have given an introduction to the theory of topological preordered spaces. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 21/22 Conclusions • We have given an introduction to the theory of topological preordered spaces. • This theory has many applications and is mathematically quite natural (quasi-uniformities). Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 21/22 Conclusions • We have given an introduction to the theory of topological preordered spaces. • This theory has many applications and is mathematically quite natural (quasi-uniformities). • The main message seems to be: Order and topology are two aspects of the same object and should be studied jointly. We expect that it will be central for a formulation of a quantum spacetime theory. Castiglioncello, September 19, 2012 Topological ordered spaces as a foundation. . . 21/22