Point-Free Topology. Erik Palmgren Uppsala University, Dept. of Mathematics

Transcription

1 Point-Free Topology Erik Palmgren Uppsala University, Dept. of Mathematics MAP Summer School 2008, International Centre for Theoretical Physics. Trieste, August 11 22, / 130

2 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Topology and choice principles It is well-known that some basic theorems classical topology use (and require) the full Axiom of Choice: Tychonov s Theorem (AC): If (X i ) i I is a family of (covering) compact spaces then the product topology is compact. Special case: The Cantor space C = {0, 1} N is compact. i I X i 2 / 130

3 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Constructive topology To carry out topology on a constructive foundation it is necessary to come to grips with compactness. Brouwer s solution: The compactness of the Cantor space follows from the nature of choice sequences. This the Fan Theorem. Moreover, it implies that the interval [0, 1] is compact. Kleene: The Cantor space is not compact under a recursive realizability interpretation. E. Bishop: Restricting to metric spaces, covering compactness should be replaced by total boundedness (Foundations of Constructive Analysis, 1967). 3 / 130

4 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Point-free topology A quite early idea in topology: study spaces in terms of the relation between the open sets. (Wallman 1938, Menger 1940, McKinsey & Tarski 1944, Ehresmann, Benabou, Papert, Isbell,...) For a topological space X the frame of open sets (O(X ), ) is a complete lattice satisfying an infinite distributive law U i I V i = i I U V i (or, equivalently, is a complete Heyting algebra). The inverse mapping of a continuous function f : X Y gives rise to a lattice morphism f 1 : O(Y ) O(X ) which preserves arbitrary suprema. 4 / 130

5 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Many important properties of topological spaces may be expressed without referring to the points, but only referring to the relation between the open sets. For example a topological space X is compact if for any family of open sets U i (i I ) in X we have if i I U i = X, then there are i 1,..., i n I so that U i1 U in = X That is, every open cover of the space has a finitely enumerable (f.e.) subcover. 5 / 130

6 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Recall: A lattice is a partially ordered set (A, ) where every finite list a 1,..., a n of elements has a supremum a 1 a n = i {1,...,n} a i and an infimum a 1 a n = i {1,...,n} a i For n = 0 these are respectively (the smallest element) and T (the largest element). 6 / 130

7 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A lattice (A, ) is a distributive if a (b c) = (a b) (a c) (Exercise show that a (b c) = (a b) (a c) follows.) A lattice A is complete if sup S exists for every subset S A. Proposition: In a complete lattice A the infimum of S A, is given by S = {x A : x is a lower bound of S}. 7 / 130

8 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Def A frame (or locale) is a complete lattice A which satisfies the infinite distributive law: a ( i I b i ) = i I a b i for any subset {b i : i I } A. Example If X is a topological space, then its open sets (O(X ), ) ordered by inclusion form a frame. Here U V = U V U i. i I U i = i I However, i I U i = Interior( i I U i ). 8 / 130

9 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A lattice A is a Heyting-algebra if there is a binary operation ( ) : A A A so that for all a, b, c A: (Cf. laws for implication.) a b c = a (b c). Theorem. Let A be a complete lattice. Then A is Heyting-algebra iff A is a frame. To prove the if-direction define (b c) = {a A : a b c}. 9 / 130

10 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies In a Heyting algebra we can define a pseudo-complement We have a = (a ). a a =. However a a = T is in general false: Example Let R be the real line with the usual topology. Then for an open subset U R we have U = {V O(R) : V U = }. In particular, (0, 1) = (, 0) (1, ) so (0, 1) (0, 1) R = T. 10 / 130

11 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A frame morphism h : A B from between frames is a lattice morphism that preserves infinite suprema: (i) h(t) = T, (ii) h(a b) = h(a) h(b), (iii) h( i I a i) = i I h(a i). The frames and frame morphisms form a category, Frm. Example A typical frame morphism is the pre-image operation f 1 : O(Y ) O(X ), where f : X Y is any continuous function. 11 / 130

12 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies For conceptual reasons one considers the opposite category of the category of frames: Def Let A and B be two locales. A locale morphism f : A B is a frame morphism f : B A. The composition g f of locale morphisms f : A B and g : B C is given by (g f ) = f g. Denote the category of locales and locale morphisms by Loc. The locale (frame) (O({ }), ) that comes from the one point space { } is denoted 1. It is the terminal object of Loc. Here T = { } and =. 12 / 130

13 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Since the points of an ordinary topological space X are in 1-1 correspondence with the maps { } X (necessarily continuous) we may define what a point of locale is by analogy: Def. Let A be a locale. A point of A is any locale morphism p : 1 A. Note that such point is a frame morphism p : A 1, which is completely determined by the elements F p = {a A : p (a) = T} sent to T. Thus we may identify points with completely prime filters F on A, i.e. F A containing T such that a F, a b = b F, a, b F = a b F, i I a i F = ( i I ) a i F. 13 / 130

14 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Fundamental adjunction Theorem The functor Ω : Top Loc is left adjoint to the functor Pt : Loc Top, that is, there is an bijection θ X,A : Loc(Ω(X ), A) = Top(X, Pt(A)), natural in X and A. Here Ω(X ) = (O(X ), ) and Ω(f ) = f 1 : Ω(Y ) Ω(X ) for f : X Y. Pt(A) is the set of points of the locale A, and Pt(f )(x) = f x. 14 / 130

15 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A locale is spatial if it has enough points to distinguish elements of the locale i.e. a = b = a = b. A space X is sober if every irreducible closed set is the closure of a unique point. (A nonempty closed C is irreducible, if for any closed C, C : C C C implies C C or C C.) The adjunction induces an equivalence between the category of sober spaces and category of spatial locales: Sob Spa Remark Ω gives a full and faithful embedding of all Hausdorff spaces into locales (in fact of all sober spaces). 15 / 130

16 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Locale Theory Formal Topology Standard Locale Theory can be dealt with in an impredicative constructive setting, e.g. in a topos (Joyal and Tierney 1984). The predicativity requirements of BISH requires a good representation of locales, formal topologies. They were introduced by Martin-Löf and Sambin (Sambin 1987) for this purpose. (A,, ) is a formal topology if (A, ) is a preorder, and is an abstract cover relation which extends. It represents a locale (Sat(A), ) whose elements are the saturated subsets U A, i.e. a U = a U (A formal topology corresponds to Grothendieck s notion of a site on a preordered set. ) 16 / 130

17 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies More in detail: A formal topology X is a pre-ordered set (X, ) of so-called basic neighbourhoods. This is equipped with a covering relation a U between elements a of X and subsets U X and so that {a X : a U} is a subset of X. The cover relation is supposed to satisfy the following conditions: (Ext) If a b, then a {b}, (Refl) If a U, then a U, (Trans) If a U and U V, then a V, (Loc) If a U and a V, then a U V. Here U V is an abbreviation for ( x U) x V. Moreover U V is short for the formal intersection U V, where W = {x X : ( y W )x y}, the down set of W. 17 / 130

18 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A point of a formal topology X = (X,, ) is a subset α X such that (i) α is inhabited, (ii) if a, b α, then for some c α with c a and c b, (iii) if a α and a b, then b α, (iv) if a α and a U, then b α for some b U. The collection of points in X is denoted Pt(X ). It has a point-set topology given by the open neighbourhoods: a = {α Pt(X ) : a α}. 18 / 130

19 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies We shall often consider different cover relations,,,... on one and the same underlying pre-ordered set (X, ). There is an natural ordering of cover relations which holds iff for all a X and for all subsets U of X, a U implies a U. We say that is smaller than. On a given preorder (X, ) there is a smallest cover relation given by a s U def a U. (Check this as an exercise.) If is = on X, this correspond to (X,, s ) being the discrete topology. There is also a largest cover relation a t U def true, which gives the trivial topology. 19 / 130

20 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Many properties of the formal topology X can now be defined in terms of the cover directly. We say that X is compact if for any subset U X For U X define X U = ( f.e. U 0 U) X U 0. U = {x X : {x} U }, the open complement of U. It is easily checked that U U, and if V U, then V U. A basic neighbourhood a is well inside another neighbourhood b if X {a} {b}. In this case we write a b. A formal topology X is regular if its cover relation satisfies a {b X : b a}. (Compact Hausdorff = Compact regular.) 20 / 130

21 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies For motivation we start with the extended example of the Cantor space. Let S be a set. Denote by S <ω the set of finite sequence of elements in S: a 1,..., a n. Concatenation of sequences u, v is denoted u v. Order the sequences by saying that w u iff w = u v for some v. Denote by S ω the set of infinite sequences of elements in S, i.e. of functions N S. We define a topology on S ω by declaring the basic open sets to be B a1,...,a n = {x S ω : ( i = 1,..., n) x(i 1) = a i }. We have B = S ω. 21 / 130

22 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies By letting S = {0, 1} we get the so-called Cantor space C = {0, 1} ω of infinite binary sequences. By letting S = N we obtain the Baire space N ω. Classically we can show that S ω is compact iff S is finite. This uses König s lemma. Brouwer s Fan Theorem implies that the Cantor space, and in fact every subspace of the form B v, is covering compact: For any subset M {0, 1} <ω : u M B u = B v = ( f.e. F M) u F B u = B v. However this theorem is really an axiom as it is false under many natural constructive interpretations, for instance recursive realizability interpretations. 22 / 130

23 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies For a finite F = {u 1,..., u m } {0, 1} <ω the covering relation u F B u = B v can be checked by considering sequences of length n = max( u 1,..., u m ). We have B u = B v ( w {0, 1} <ω )( w = n, w v w F ). u F We abbreviate the righthand side as the property K(v, n, F ). This property is clearly decidable given v, n and F. 23 / 130

24 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies We shall now define a formal topology corresponding to the Cantor space. Let C = ({0, 1} <ω,, ) where is as above and is given by v U def ( f.e. F U)( n) K(v, n, F ). Now compactness is built-in to the cover relation. Thm. C is a compact formal topology and Pt(C) = C. Note: for any u u {u 0, u 1 }. 24 / 130

25 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies The formal topology C = (C,, ) is inductively generated by the following axiom in the sense that u {u 0, u 1 } Thm If is another cover relation so that (C,, ) is a formal topology and then u {u 0, u 1 } (u C) a U = a U. This gives an induction principle for the cover relation which is an important proof method. 25 / 130

26 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies For a given preorder (X, ) an cover axiom is a pair (a, G) where a X and G is a subset of X. We suggestively write this pair as a G. A formal topology X = (X,, ) is (inductively) generated by a family {a i G i } i I of cover axioms, if is the smallest cover relation on (X, ) which satisfies all the axioms, i.e. a i G i for all i I. 26 / 130

27 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies We may define the formal Baire space as B = (N <ω,, ) where is is inductively generated by u {u n : n N} (u N <ω ) Proposition: B is a regular formal topology. Proof: We have u u, since N <ω u u. NB: Such covers can be shown to exist using strong induction axioms. In fact for any set of cover axiom, there is a cover relation generated by them. 27 / 130

28 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Set-presented vs inductively generated formal topologies Def. A formal topology X = (X,, ) is set-presented if there is a family of subsets C(a, i) X (a X, i I (a)) so that a U ( i I (a))c(a, i) U. Example 1. Impredicatively, every formal topology X is set-presented by I (a) = {U X : a U} and C(a, U) = U. 2.The Cantor space is set-presented by I (a) = {(F, n) P f.e. (X ) N : K(a, n, F )} C(a, (F, n)) = F Theorem. A formal topology is set-presented iff it is inductively generated by some set of cover-axioms. 28 / 130

29 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Induction on covers As a simple illustration of the method of induction on covers we give alternative characterisation of points in an inductively generated formal topology A = (A,, ). Suppose that A is generated by the axioms R = {a i V i } i I. We call α A an R-splitting filter, if it satisfies conditions (i)-(iii) for a point but instead of (iv), only the more restrictive condition that for any i I (This is often easier to check.) a i α = ( b V i ) b α. (1) 29 / 130

30 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Induction on covers (cont.) Proposition. Each R-splitting filter is a point. Proof. We need to prove that for each R-splitting filter α, the condition (iv) holds. The trivial but crucial observation is that (iv) can be expressed as: a U = [a α ( b U)b α]. Since is the smallest cover relation satisfying R it is now sufficient to prove that a U def [a α ( b U)b α]. defines a cover relation satisfying R. By (1) it clearly satisfies R. 30 / 130

31 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Induction on covers (cont.) It remains to check that satisfies (Ext), (Refl), (Trans) and (Loc). (Ref) is immediate. (Ext) follows by (ii). (Loc) follows by (iii). (Trans): Suppose a U and U W. We want to show a W : suppose a α. Hence there is b U with b α. Thus b W, and hence ( c W )c α, which was to be shown. For an inductively generated formal topology A, the collection of points Pt(A) is in generally only a class. 31 / 130

32 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies A point-wise cover which is not a formal cover In realizability models of intuitionistic systems the so-called Church Thesis (better named Church-Turing Thesis) is valid. It says that every function on natural numbers is computable by a Turing machine: (CT) ( f : N N) ( e N)( x N)( y N)(T (e, x, y) U(y) = f (x)). Here T is the Kleene predicate and U is the result extracting function. A consequence is e.g. that the number of points in the Cantor space (and similar spaces) will be restricted in such models. The strategy is ito find cover which just covers those points, but is still not a formal (inductive) cover. 32 / 130

33 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Assuming (CT) we shall prove that there is a set of basic opens V C so that C = v V B v but C V fails. Let where V = V 0 V 1, V k = { a 0,..., a n 1 : ( i, j < n)(a i = k T (i, i, j) U(j) = k)}. Note that V k is decidable and (V k ) V k. To prove C v V B v, take an arbitrary f C. We show f B v for some v V. Let e be the Kleene index for f, i.e. f (n) = k ( y)t (e, n, y) U(y) = k. (2) We evaluate f (e). Suppose T (e, e, u). We have f B v for v = f (0),... f (e + u). (Trivial.) For f (u) = k {0, 1} we have U(u) = k by (2). Then by definition v V k, so v V as required. 33 / 130

34 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Suppose C V. By definition there is a finite F V and m so that w F for every w with w m. (In particular C F.) We show this is impossible. Define a finite sequence w = d 0,..., d m 1 C by letting d i = 1 def i < m ( j < m)(t (i, i, j) U(j) = 0). (3) Then w v for some v F. Since F V = V we have w V = V 0 V 1. Suppose w V 0. By (2) there are i, j < m with d i = 0, T (i, i, j) and U(j) = 0. According to (3) this implies d i = 1 which is a contradiction. Thus w V 1. Hence there are i, j < m with d i = 1, T (i, i, j) and U(j) = 1. But by (3) it follows that U(j) = 0. A contradiction again. Hence C V is false. 34 / 130

35 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies The example is a straightforward adaptation of a theorem which shows that the Fan Theorem is incompatible with CT. The original result is due to Kleene. This example shows that the Fundamental Adjunction Ω Pt : Top Loc cannot be used without further ado to transfer results from spaces to locales in a constructive setting. It is necessary to develop point-free techniques to obtain constructive results. 35 / 130

36 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies Point-free proofs are more basic Theorem (Johnstone 1981): Tychonov s Theorem holds for locales, without assuming AC. Slogan of B. Banaschewski: choice-free localic argument + suitable choice principles = classical result on spaces In fact, Tychonov s theorem for locales is constructive even in the predicative sense (Coquand 1992). 36 / 130

37 Non-constructive aspects of topology Locales Formal Topology Inductively generated formal topologies This suggestive slogan can be generalised constructive localic argument + Brouwerian principle = intuitionistic result on spaces. However, since the work of the Bishop school (BISH) on constructive analysis it is known that there is often a basic BISH constructive argument + Brouwerian principle = intuitionistic result on metric spaces. A natural question: how does BISH constructive topology and constructive locale theory relate? For instance on metric spaces. 37 / 130

38 Real numbers Subspaces Category of formal topologies Formal reals R The basic neighbourhoods of R are {(a, b) Q 2 : a < b} given the inclusion order (as intervals), denoted by. The cover is generated by (G1) (a, b) {(a, b ) : a < a < b < b} for all a < b, (G2) (a, b) {(a, c), (d, b)} for all a < d < c < b. The set of points Pt(R) of R form a structure isomorphic to the Cauchy reals R. For a point α with (a, b) α we have by (G2) e.g. (a, (a + 2b)/3) α or ((2a + b)/3, b) α. 38 / 130

39 Real numbers Subspaces Category of formal topologies G1: ( ). ( ). ( ) G2: ( ) ( ) ( ) 39 / 130

40 Real numbers Subspaces Category of formal topologies An elementary characterisation of the cover relation on formal reals is: (a, b) U ( a, b Q)(a < a < b < b ( finite F U) (a, b ) F ) Note that for finite F the pointwise inclusion (a, b ) F is decidable, since the end points of the intervals are rational numbers. (Exercise) This characterization is crucial in the proof of the Heine-Borel theorem (see Cederquist, Coquand and Negri). 40 / 130

41 Real numbers Subspaces Category of formal topologies For any rational number q there is a point in Pt(R); ˆq = {(a, b) Q 2 : a < q < b}. For a Cauchy sequence x = (x n ) of rational numbers we define ˆx = {(a, b) Q 2 : ( k)( n k) a < x n < b} which is a point of R. Order relations on real numbers α, β Pt(R): α < β def ( (a, a ) α)( (b, b ) β) a < b α β def (β < α) ( (b, b ) β)( (a, a ) α)a b 41 / 130

42 Closed subspaces Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies Let X be a formal topology. Each subset V X determines a closed sublocale X \ V whose covering relation is given by a U def a U V. Note that V and V, so the new cover relation identifies the open set V with the empty set. What remains is the complement of V. Theorem If X is compact, and U X a set of neighbourhoods, then X \ U is compact. 42 / 130

43 Real numbers Subspaces Category of formal topologies We define he unit interval by [0, 1] = R \ V where V = {(a, b) : b < 0 or 1 < a}. Denote the cover relation of [0, 1] by. Heine-Borel theorem [0, 1] is a compact formal topology. Proof. Suppose X U, where X is the basic nbhds of R. Thus in particular (0 2ε, 1 + 2ε) U and thus (0 2ε, 1 + 2ε) V U. By the elementary characterisation of there is a finite F V U with (0 ε, 1 + ε) F. As F is finite, we can prove (0 ε, 1 + ε) F using G1 and G2. Also we find finite F 1 F U U so that (0 ε, 1 + ε) V F 1. Hence (0 ε, 1 + ε) F 1. But X (0 ε, 1 + ε) so X F / 130

44 Open subspaces Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies Let X be a formal topology. Each set V X of neighbourhoods determines an open sublocale X V whose covering relation is given by a U def a V U. Note that U 1 U 2 iff U 1 V U 2 V. Hence only the part inside V counts when comparing two open sets. Example For two points α, β R the open interval (α, β) is R V where V = {(a, b) : α < â & ˆb < β}. 44 / 130

45 Real numbers Subspaces Category of formal topologies Continuous maps relate the covers Let X = (X,, ) and Y = (Y,, ) be formal topologies. A relation F X Y is a continuous mapping X Y if U V = F 1 U F 1 V, ( preservation of arbitrary sups ) X F 1 Y, ( preservation of finite infs ) a F 1 V, a F 1 W = a F 1 (V W ). a U, x F b for all x U = a F b, Each such induces a continuous point function f = Pt(F ) given by α {b : ( a α)r(a, b)} : Pt(X ) Pt(Y) and that satisfies: a F b f [a ] b. 45 / 130

46 The category of formal topologies. Real numbers Subspaces Category of formal topologies The inductively generated formal topologies and continuous mappings form a category FTop, which is classically equivalent to Loc. They both share many abstract properties with the category of topological spaces which can be expressed in category theoretic language. This makes it possible to describe topological constructions without mentioning points. 46 / 130

47 Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies Recall: A product of objects A and B in a category C is an object A B and two arrows π 1 : A B A and π 2 : A B B in C, so that for any arrows f : P A and g : P B there is a unique arrow f, g : P A B so that π 1 f, g = f and π 2 f, g = g. π 1 π 2 A A B f B f,g g c 47 / 130

48 Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies The dual notion of product is coproduct or sum. A coproduct of objects A and B in a category C is an object A + B and two arrows ι 1 : A A + B and ι 2 : B A + B in C, so that for any arrows f : A P and g : B P there is a unique arrow ( f g ) : A + B P so that ( f g ι 1 ι 2 A A + B f B ( f g) g c ) ι1 = f and ( f g) ι2 = g. 48 / 130

49 Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies A equalizer of a pair of arrows A f B in a category C is a object E and an arrow e : E A with fe = ge so that for any arrow k : K A with fk = gk there is a unique arrow t : K E with et = k. g E t K e k A In Top: E = {x A : f (x) = g(x)}. Dual notion: A coequalizer of a pair of arrows A f B in a category C is a object Q and an arrow q : B Q with qf = qg so that for any arrow k : B Q with kf = kg there is a unique arrow t : Q K with tq = k. 49 / 130 f g B g

50 Real numbers Subspaces Category of formal topologies Categorical topology. As the category of topological spaces has limits and colimits, many spaces of interest can be built up using these universal constructions, starting from the real line and intervals. The circle {(x, y) R R : x 2 + y 2 = 1} is an equaliser of the constant 1 map and (x, y) x 2 + y 2. It can also be constructed as a coequaliser of s, t : { } [0, 1] where s( ) = 0, t( ) = 1. (Identifying ends of a compact interval.) 50 / 130

51 Real numbers Subspaces Category of formal topologies The categorical properties of the category FTop of set-presented formal topologies ought to be same as that of the category of locales Loc. Theorem. Loc has small limits and small colimits. However, since we are working under the restraint of predicativity (as when the meta-theory is Martin-Löf type theory) this is far from obvious. (Locales are complete lattices with an infinite distributive law.) 51 / 130

52 Real numbers Subspaces Category of formal topologies FTop has... Limits: Products Equalisers (and hence) Pullbacks Colimits: Coproducts (Sums) Coequalisers (and hence) Pushouts Moreover: certain exponentials (function spaces): X I, when I is locally compact. 52 / 130

53 Real numbers Subspaces Category of formal topologies Further constructions using limits and colimits 1. The torus may be constructed as the coequaliser of the followings maps R 2 Z 2 R 2 (x, n) x (x, n) x + n. 2. The real projective space RP n may be constructed as coequaliser of two maps R n+1 R 0 R n+1 (x, λ) x (x, λ) λx. 53 / 130

54 Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies 3. For A X and f : A Y, the pushout gives the attaching map construction: A f Y X Y f X 4. The special case of 3, where Y = 1 is the one point space, gives the space X /A where A in X is collapsed to a point. 54 / 130

55 Products of formal topologies Real numbers Subspaces Category of formal topologies Let A = (A, A, A ) and B = (B, B, B ) be inductively generated formal topologies. The product topology is A B = (A B,, ) where (a, b) (c, d) def a A c & b B d and is the smallest cover relation on (A B, A B ) so that a A U = (a, b) U {b}, b B V = (a, b) {a} V. The projection π 1 : A B A is defined by (Second projection is similar.) (a, b) π 1 c def (a, b) A {c} 55 / 130

56 Real numbers Subspaces Category of formal topologies Example The formal real plane R 2 = R R has by this construction formal rectangles ((a, b), (c, d)) with rational vertices for basic neighbourhoods (ordered by inclusion). The cover relation may be characterized in a non-inductive way as ((a, b), (c, d)) U ( u, v, x, y)[a < u < v < b, c < x < y < d ( finite F U)(u, v) (x, y) F ] 56 / 130

57 Formal topological manifolds Real numbers Subspaces Category of formal topologies A formal topology M is an n-dimensional manifold if there are two families of sets of open neighbourhoods {U i M} i I and {V i R n } i I and a family of isomorphisms in FTop ϕ i : M Ui (R n ) Vi (i I ) so that M {U i : i I }. 57 / 130

58 Part 1: Locales and formal topology Subspaces defined by inequations Real numbers Subspaces Category of formal topologies Let V R 2 be the set of open neighbourhoods aboove the graph y = x i.e. V = {((a, b), (c, d)) R 2 : b < c} Then L = (R 2 ) V R 2 has for points pairs (α, β) with α < β. Let f, g : X R be continuous maps. Then form the pullback: S X L R 2 f,g Then the open subspace S X has for points those ξ Pt(X ) where f (ξ) < g(ξ). 58 / 130

59 Real numbers Subspaces Category of formal topologies Subspaces defined by inequations (cont.) Similarly we may define subspaces by non-strict inequalities. Define K = (R 2 \ V ) R 2, then K has for points (α, β) R 2 such that α β. Replacing L by K is the previous pullback diagram we get that S X has for points those ξ Pt(X ) where f (ξ) g(ξ). 59 / 130

60 Coequalizers in formal topology Real numbers Subspaces Category of formal topologies Coequalizers are used to built quotient spaces, and together with sums, to attach spaces to spaces. As the category Loc is opposite of the category Frm the coequalizer of A f B in Loc can be constructed as the equalizer g of the pair B f A in Frm. g As Frm is algebraic the equalizer can be constructed as E = {b B : f (b) = g (b)} B f A g 60 / 130

61 Real numbers Subspaces Category of formal topologies Coequalizers in formal topology (cont.) A direct translation into FTop yields the following suggestion for a construction: {U P(B) : F 1 U = G 1 U} for a pair of continuous mappings F, G : A topologies. Here W = {a A : a W }. B between formal Problem: the new basic neighbourhoods U do not form a set. 61 / 130

62 Real numbers Subspaces Category of formal topologies It turns out that can one find, depending on F and G, a set of subsets R(B) with the property that F 1 U = G 1 U, b U = ( V R(B))(b V U & F 1 V = G 1 V ) Now the formal topology, whose basic neighbourhoods are Q = {V R(B) : F 1 V = G 1 V }, and where U Q W iff U B W for W R(B), defines a coequaliser. Moreover the coequalising morphism P : B Q is given by: a P U iff a B U. 62 / 130

63 Real numbers Subspaces Category of formal topologies Cohesiveness of formal spaces: joining paths Def The space Y has the Path Joining Principle (PJP) if f : [a, b] Y and g : [b, c] Y are continuous functions with f (b) = g(b), then there exists a unique continuous function h : [a, c] Y, with h(t) = f (t) for t [a, b], and h(t) = g(t) for t [b, c]. Theorem Each complete metric space Y satisfies PJP. 63 / 130

64 Real numbers Subspaces Category of formal topologies For general metric spaces PJP fails constructively: Proposition There is a metric space Y (=[ 1, 0] [0, 1]), such that if the PJP is valid for Y, then for any real x x 0 or x 0. However: Proposition In FTop the PJP holds for any Y. (Useful for defining fundamental groups.) 64 / 130

65 Part 1: Locales and formal topology Real numbers Subspaces Category of formal topologies There is a stronger version (HJP) which allows for the composition of homotopies: Proposition: Let X be a formal topology. For α β γ in Pt(R), the diagram 1 X, ˆβ X X [β, γ] 1 X, ˆβ 1 X E 2 X [α, β] X [α, γ] 1 X E 1 (4) is a pushout diagram in FTop. Here E 1 and E 2 are the obvious embeddings of subspaces. 65 / 130

66 Real numbers Subspaces Category of formal topologies The following lemma is used in the proof. It may be regarded as a (harmless) point-free version of the trichotomy principle for real numbers. Lemma For any point β of the formal reals R let T β = {(a, b) R : b < β or (a, b) β or β < a}. Then for any U R we have U R U T β. Here V R W means that V R W and W R V. 66 / 130

67 Real numbers Subspaces Category of formal topologies Proof. The covering U T β R U is clear by the axioms (Tra) and (Ext). To prove the converse covering, suppose that (a, b) U. Then it suffices by axiom (G1) to show (c, d) R U T β for any a < c < d < b. For any such c, d we have β < c or a < β, by the co-transitivity principle for real numbers. In the former case, (c, d) U T β, so suppose a < β. Similarly comparing d, b to β we get d < β, in which case (c, d) U T β, or we get β < b. In the latter case we have (a, b) U T β, so in particular (c, d) U T β. 67 / 130

68 Locally compact metric spaces in BISH Def (Bishop) An inhabited metric space X is locally compact, if each bounded subset can be included in some compact subset of X. Prop Any locally compact space is complete and separable. Rem Q and (0, 1) are not locally compact in the above sense. Def A function f : X Y between a locally compact metric space X and a metric space Y is continuous, if f is uniformly continuous on each compact subset of X. 68 / 130

69 Constructivity of the Fundamental Adjunction? By the adjunction Ω : Top Loc Pt : Loc Top the functor Ω gives a full and faithful embedding of all Hausdorff spaces into locales. However, the functor Ω yields only point-wise covers in Loc, so Ω({0, 1} ω ) is not compact, (unless e.g. the Fan Theorem is assumed). The localic version of the Cantor is compact, as we have seen. To embed the locally compact metric spaces used in BISH we need a more refined functor. 69 / 130

70 Localic completion (S. Vickers) For any metric space (X, d) its localic completion M = M X is a formal topology (M, M, M ) where M is the set of formal ball symbols {b(x, δ) : x X, δ Q + }. These symbols are ordered by formal inclusion b(x, δ) M b(y, ε) d(x, y) + δ ε b(x, δ) < M b(y, ε) d(x, y) + δ < ε. 70 / 130

71 71 / 130

72 The cover is generated by (M1) p {q M : q < p}, for any p M, (M2) M {b(x, δ) : x X } for any δ Q +, The points of M X form a metric completion of X. There is a metric embedding j X = j : X Pt(M X ) given by j(u) = {b(y, δ) : d(u, y) < δ}. Moreover, j X is an isometry in case X is already complete. For X = Q, we get the formal real numbers modulo notation for intervals b(x, δ) = (x δ, x + δ). 72 / 130

73 Remarks Countable choice is used to prove the isometry- The covering relation is in the case when X is the Baire space is given by an infinite wellfounded tree. Must allow generalised inductive definitions in the meta theory. 73 / 130

74 Theorem (Vickers) Let X be a complete metric space. Then: X is compact iff M X is compact. Theorem 2 (P.) If X = (X, d) is locally compact, then M X is locally compact as a formal topology, i.e. p {q M : q p} for any p. Here the way below relation is q p iff for every U: p U implies there is some finitely enumerable (f.e.) U 0 U with q U / 130

75 Elementary characterisation of the cover of M X Refined cover relations U V df ( p U)( q V )p q Note: If U V, then U V. (Similar extension for <) p ε U df ( q p)[radius(q) ε {q} U] Write p U iff p ε U for some ε Q +. Note: If p V, then p V. 75 / 130

76 Any sufficiently small disc included in the inner left disc slips into one of the dashed discs. 76 / 130

77 Elementary characterisation of the cover of M X Lemma 4 Let X be locally compact. Let δ Q +. For any formal balls p < q there is a finitely enumerable C such that p C < q where all balls in C have radius < δ. Remark: In fact, this is as well a sufficient condition for local compactness. Let A(p, q) = {C P fin.enum. (M X ) : p C < q}. 77 / 130

78 Elementary characterisation of the cover of M X The following cover relation is generalising the one introduced by Vermeulen and Coquand for R: a U df ( b < c < a) ( U 0 A(b, c)) U 0 < U. Theorem 3 If X is a locally compact metric space, then a U a U ( holds for any metric X space.) Proof of goes by verifying that is a cover relation satisfying (M1) and (M2). 78 / 130

79 Let f : X Y be a function between complete metric spaces. Define the relation A f M X M Y by Here b(x, δ) = B(x, δ). a A f b df a {p : ( q < b)f [p ] q } Lemma 5 A f : M X M Y is a continuous morphism between formal topologies, whenever X is locally compact and f is continuous. Define thus M(f ) : M X M Y as A f Remark Classically, a A f b is equivalent to f [a ] b. However, this definition does not work constructively. 79 / 130

80 Embedding Theorem Embedding Theorem: The functor M : LCMet FTop is a full and faithful functor from the category of locally compact metric spaces to the category of formal topologies. It preserves finite products. Explanation: Suppose that X and Y are locally compact metric spaces. M faithful means: For f, g : X Y continuous functions f = g M(f ) = M(g) M full means: any continuous mapping F : M(X ) M(Y ) is the comes from some (unique) continuous function f : X Y : F = M(f ). 80 / 130

81 That M preserves binary products means e.g. that if X π 1 X Y π 2 Y is a product diagram, then so is: M(X ) M(π 1 ) M(X Y ) M(π 2 ) M(Y ). Hence M(X Y ) = M(X ) M(Y ). Since we have M(R) = R we may then lift any continuous operation f : R n R from Cauchy reals to formal reals M(f ) : R n R. 81 / 130

82 Theorem For F, G : R n R and f = Pt(F), g = Pt(G) : R n R we have F G ( x)f ( x) g( x). Here F G is defined via factorization through a closed sublocale, as follows: For F, G : A R continuous we say that F G iff G, F : A R 2 factorizes through R 2 \ V R 2 where V = {((a, b), (c, d)) R 2 : b < c}. 82 / 130

83 There is a characterisation of maps into closed subspaces. For a set of neighbourhoods W Y let Y \ W denote the corresponding closed sublocale, and let E W : Y \ W Y be the embedding. Let F : X Y be a continuous morphism. TFAE: F factors through E W. F 1 W X. F : X (Y \ W ) is continuous. Def F : X R is non-negative if it factorises through R \ N, where N is the set of b(x, δ) with x < δ. 83 / 130

84 The previous theorem may be used prove equalities and inequalities ( ) for formal real numbers by standard point-wise proofs. Results of constructive analysis are readily reused. More generally we have Theorem Let X be a locally compact metric space. For F, G : M(X ) R and f = Pt(F), g = Pt(G) : X R we have F G ( x X )f (x) g(x). 84 / 130

85 It would also be desirable to reuse results about strict inequalities (<). There is however no such straightforward transfer theorem as for. This is connected to the problem of localising infima of continuous functions [0, 1] R, which we consider as a preparation. 85 / 130

86 The infimum problem on compact sets Problem: In BISH find a positive uniform lower bound of a uniformly continuous function f : [0, 1] (0, ). This is impossible by results of Specker 1949, Julian & Richman 1984 using a recursive (computable) interpretation. (Classically, f attains its minimum.) However in Brouwerian intuitionism (INT), a uniform lower bound can be found using the FAN theorem. 86 / 130

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88 In Bishop s constructive analysis this phenomenon is serious problem. ( P. Schuster: What is continuity, constructively?) We ought to have (*) The composition of two continuous functions is continuous. Consider, however, the composition of f with the reciprocal: [0, 1] f (0, ) 1/( ) R. If 1/f is uniformly continuous, we can find an upper bound of the function on the compact the compact interval [0, 1]. But this means that we can also find a uniform lower bound of f. One concludes that (*) fails. The two versions of continuity are incompatible. Point-free topology has a solution. 88 / 130

89 Open subspaces X G the open subspace of X given by a set of formal nbds G. E G : X G X the associated embedding. Lemma Let F : X Y be a continuous morphism between formal topologies. Let G Y be a set of nbds. TFAE: F factorises through E G : Y G Y X F 1 [G] F : X Y G is continuous. Def F : X R is positive if it factorises through R P, where P is the set of b(x, δ) with x > δ. The above Lemma gives several ways to prove positivity. 89 / 130

90 Locally uniformly positive maps Let X be a metric space. A function f : X R is locally uniformly positive (l.u.p) if for every x X and every δ > 0 there is some ε > 0 so that for all y Y d(y, x) < δ = f (y) > ε. Theorem Let X be a locally compact metric space, and suppose f : X R is continuous. Then f is l.u.p. if, and only if, M(f ) : M(X ) R is positive. Corollary M(f ) < M(g) g f is l.u.p. 90 / 130

91 Proof of Theorem ( ): Suppose M(f ) is positive. Using Theorem 3 we obtain M(X ) M(f ) 1 [G]. Let x X and δ > 0. Then consider neighbourhoods s = b(x, δ) < s < p. Thus we have some p 1,..., p n < s with s {p 1,..., p n } < M(f ) 1 [G] Thus there are q i > p i and r i G, i = 1,..., n, satisfying f [(q i ) ] (r i ). Thereby f [s ] = f [b(x, δ) ] f [p 1] f [p n] (r 1 ) (r n ). From which follows f [B(x, δ)] = f [b(x, δ) ] (ε, ), so f is indeed l.u.p. 91 / 130

92 Continuous maps between open subspaces (Work in progress) Let X and Y be locally compact metric spaces. For open subsets U X and V Y a suitable definition of continuous map U V in BISH is: f : U V is continuous iff for every totally bounded S U, C1) f is uniformly continuous on S and C2) f [S] V. Here S U def ( t > 0)({x X : d(x, S) t} U). 92 / 130

93 Note that the meaning of is strongly dependent on the underlying space X : Write S r for {x X : d(x, S) t}. X = [0, 1]: Then S = [0, 1] is totally bounded and S r = [0, 1] for any r. Hence S X. For Y = V = R, we get the standard notion of (uniform) continuity: [0, 1] R. For Y = R and V = (0, + ), we get uniformly continuous functions which are uniformly positive. X = R, U = (0, + ): Note that S = (0, 1) is totally bounded, but we do not have S U. The reciprocal will be continuous (0, + ) R. 93 / 130

94 Thm Let X and Y be locally compact metric spaces. For formal open (and saturated) A M X and B M Y, a continuous morphism F : (M X ) A (M Y ) B induces a point function f : A B which satisfies the following conditions: for every totally bounded S < A K1) f is uniformly continuous on S and K2) f [S] < B. Here S < A def ( f.e. F M X ) S F & F < A. 94 / 130

95 The continuity conditions K1-2 generalize the standard C1-2. For any open U X, there is a largest formal open saturated A(U) = {a M X : a U}. We have Lemma: S < A(U) S U. We have a converse to the previous theorem. Thm If f : A B is K-continuous then A f : (M X ) A (M Y ) B is a continuous mapping. 95 / 130

96 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Recall that a metric space X is totally bounded if for any ε > 0 there is a nonempty finitely enumerable subset {x 1,..., x n } X (called an ε-net) so that X B(x 1, ε) B(x n, ε). Here B(x, ε) = {y X : d(x, y) < ε}. The space is compact if in addition it is complete. Unlike classical topological spaces and unlike formal topologies closed subspaces of compact metric spaces are not necessarily compact. 96 / 130

97 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Brouwerian counter example: Consider the following closed subspace of [0, 1]: K = {0} {1 : P} where P is some arbitrary proposition. If K was compact, we could ask for 1/3-net of K. Say it is {x 1,..., x n }. By definition of a net, we have x k K for all k. If x k > 1/3 then x k = 1, and if x k < 2/3, then x k = 0. Checking for all k = 1,..., n, we may thus decide whether P or P. We conclude if every closed subspace of [0, 1] is compact, then PEM holds. 97 / 130

98 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Recall: for complete metric spaces X we have X is compact iff M(X ) is compact. So, taking X = [0, 1], then M = M([0, 1]) is compact. We know that M \ U is compact as a formal topology for any U M. We easily construct U so that there is an isometry Pt(M \ U) = K where K is as in the counter example. Hence Pt(M \ U) need not be compact as a metric space. 98 / 130

99 Overtness Part 1: Locales and formal topology Overt formal topologies Domains and Formal topology Predicativity and formalization issues Thus we must require something more than compactness of M \ U to guarantee that Pt(M \ U) is compact in the metric sense. The notion of overtness may be used to get a localic counterpart to Bishop s notion of compactness (P. Taylor 2000). See also T. Coquand and B. Spitters (2007) Computable sets: located and overt locales. 99 / 130

100 Overt formal topologies Domains and Formal topology Predicativity and formalization issues A formal topology X = (X,, ) is overt if there is a set P X of neighbourhoods that are positive, which means P is inhabited, a U, a P U P is inhabited, U U P. This P is necessarily unique if it exists. P is usually called a positivity predicate for X. Remark. In classical set theory, any formal topology X is overt, provided it hasa least one neighbourhood which is not covered by. We may take P = {a X : a }. The notion is thus exclusively of constructive interest. 100 / 130

101 Overt formal topologies Domains and Formal topology Predicativity and formalization issues The following is implicit in (Coquand and Spitters 2007) as explained by Coquand. Theorem Let X be a metric space and let M = M(X ) be its local completion. Let U M be such that M \ U is compact and overt. Then Pt(M \ U) is compact in the metric sense. To prove it we use: Lemma: Let U M be such that M \ U is overt. If a is a positive neighbourhood of M \ U, then α a for some α Pt(M \ U). 101 / 130

102 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Proof of Lemma: Suppose that P is the positivity predicate of M \ U. Denote the cover relation of M \ U by. Suppose a = b(x, δ) P. Let a 1 = a. Suppose we have constructed in P: a 1 a 2 a n, so that radius(a k+1 ) radius(a k )/2. By (M1) and localisation we get a n {a n } {b(y, ρ) : y X } where ρ = radius(a n )/2. Since a n P we obtain some b {a n } {b(y, ρ) : y X } with b P. Clearly radius(b) radius(a n )/2. Let a n+1 = b. 102 / 130

103 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Let α = {p M : ( n)a n p} Since the radii of a n are shrinking, this defines a point in Pt(M). (Note that we used Dependent Choice.) We claim that α Pt(M \ U) = Pt(M) \ U. Suppose that α U, i.e. for some c U: c α. Hence there is n with a n c. Thus a n U, that is a n. But since a n is positive, this is impossible! So α Pt(M \ U). 103 / 130

104 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Proof of Theorem: Let P and be as in the proof of Lemma. Let ε > 0 be given. Then by axiom M2 and positivity M {b(x, ε/2) : x X } {b(x, ε/2) : x X } P. By compactness there is some finitely enumerable F = {b(x 1, ε/2),..., b(x n, ε/2)} {b(x, ε/2) : x X } P so that M F. (5) Since each b(x i, ε/2) is positive there is by Lemma some α i b(x i, ε/2) which is in Pt(M \ U). By (5) it follows that each point in Pt(M \ U) has smaller distance than ε to some point α i. Thus {α 1,..., α n } is the required ε-net. 104 / 130

105 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Domains and Formal topology For certain purposes it is desirable to generalize the real number system to include also non-computable reals or partially determined real numbers (à la interval analysis); see Richman 1998 and also Negri We may do this for formal real numbers R by dropping one or both of the axioms (M1) p {q M : q < p}, for any p M, (M2) M {b(x, δ) : x X } for any δ Q +. If we drop both (M1) and (M2) we get R 0 whose points are non-empty and filtering sets. For instance the interval [x, y] = {(a, b) Q 2 : a < x < y < b} is a point in R / 130

106 Overt formal topologies Domains and Formal topology Predicativity and formalization issues For an arbitrary formal topology X the points Pt(X ) are partially ordered by inclusion. In many spaces of interest (e.g. regular spaces) the points α are maximal in this order, i.e. for α, β Pt(X ) α β = α = β. (If all points are maximal, then X is called a T 1 -topology.) Thm. If X is a formal topology, then (Pt(X ), ) is a directed complete partial order. Indeed the theory of formal topologies may be regarded a generalization of domain theory (Martin-Löf, Sambin). A formal topology X = (X, X, X ) is unary if a X U = ( b U) a X b. In such a topology a = {b X : a X a X. b} is a point, for any 106 / 130

107 Overt formal topologies Domains and Formal topology Predicativity and formalization issues A pre-order (X, ) is a consistently complete lower semi-lattice pre-order (CLSP) if it has a largest element T and for any consistent pair x, y X, i.e. which has a lower bound t x and t y, there is an element x y satisfying (a) and (b) above. Theorem. (Sambin, Valentini, Virgili) If X is a unary formal topology, where (X, X ) is a CLSP, then D = (Pt(X ), ) is a Scott domain. 107 / 130

108 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Define the Scott compactification X 0 of X : For a formal topology X define X0 by a X0 b a X {b} and define X0 by a X0 U ( b U)a X0 b. Let X 0 = (X, X0, X0 ). We have Theorem. If (X, ) is CLSP (a) X 0 is a unary formal topology, (b) Pt(X ) Pt(X 0 ), (c) Pt(X ) ext X0 (U) = ext X (U) Here is ext X (U) is the set of points in X who contains some neighborhood of U. 108 / 130

109 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Let F : S T be a continuous map between arbitrary formal topologies. Then the point function f = Pt(F ) : Pt(S) Pt(T ) preserves directed suprema, i.e. f ( i I α i ) = i I f (α i ) for directed sets α i (i I ) of points. 109 / 130

110 Part 1: Locales and formal topology Lifting maps from spaces to domains Overt formal topologies Domains and Formal topology Predicativity and formalization issues Theorem (P. 2007) Let F : S T be a continuous map between LSP formal topologies. Then F 0 = F is also a continuous map from S 0 to T 0 with makes this lifting square commute: F 0 S 0 T 0 H S H T S. F T Here H X : X X 0 is the natural embedding. LSP= the preorder is a lower semilattice. 110 / 130

111 Overt formal topologies Domains and Formal topology Predicativity and formalization issues Predicativity and formalization in type theory There are two main formal systems intended for the formalization of Bishop constructivisim. 1. Constructive set theory CZF which uses the ordinary set-theoretic language. A central feature is that there are no power sets and that separation ({x X : ϕ}) may only be done using restricted (bounded) formulas ϕ. 2. Martin-Löf type theory, for which a lot of computer implementations/formalizations of constructive mathematics have been done. We shortly indicate such a system MLTT which has similar features to CZF. 111 / 130