The Cantor space as a Bishop space

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1 Ludwig-Maximilians Universität Munich Fifth Workshop on Formal Topology: Spreads and Choice Sequences Institut Mittag-Leffler, Sweden

2 The history of the subject 1 The theory of Bishop spaces (TBS) is an approach to constructive point-function topology. 2 Bishop introduced function spaces, here called Bishop spaces, in Myhill commented on them in his 1975-paper. 4 Bridges revived the subject in Directly after that Ishihara studied the relation of the subcategory Fun of the category Bis of Bishop spaces with the category of neighborhood spaces Nbh (2013). 6 We try to elaborate TBS.

3 The main characteristics of TBS 1 Points are accepted from the beginning, hence it is not a point-free approach to topology. 2 Most of its notions are function-theoretic. Set-theoretic notions are avoided or play a secondary role to its development. 3 It is constructive. We work within Bishop s informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included. 4 It has simple foundation and it follows the style of standard mathematics.

4 Bishop 1973: The constructivization of general topology is impeded by two obstacles First, the classical notion of a topological space is not constructively viable. Second, even for metric spaces the classical notion of a continuous function is not constructively viable; the reason is that there is no constructive proof that a (pointwise) continuous function from a compact metric space to R is uniformly continuous.... Since uniform continuity cannot be formulated in the context of a general topological space, the latter concept also is left with no useful function to perform.

5 What if there is a constructive notion F of an abstract topological space which does not copy or follow the pattern of the classical topological space, and what if there is a constructive notion of a continuous function between two such spaces F and G such that although uniform continuity is not part of the definition of this notion, in many expected cases it is reduced to uniform continuity. Then one can hope that the problem in the constructivisation of topology posed by Bishop can be bypassed.

6 Continuity as a primitive notion A Bishop space is a pair F = (X, F ), where X is an inhabited set and F F(X ), a Bishop topology, or simply a topology, satisfies the following conditions: (BS 1 ) a R a F. (BS 2 ) f F g F f + g F. (BS 3 ) f F φ Bic(R) φ f F, (BS 4 ) f F(X ) U(F, f ) f F, If f, g F(X ), ɛ > 0, and Φ F(X ), we define U(g, f, ɛ) and U(Φ, f ) by U(g, f, ɛ) := x X ( g(x) f (x) ɛ), U(Φ, f ) := ɛ>0 g Φ (U(g, f, ɛ)). fg, λf, f, f g, f g, f F Const(X ) F F(X ) A morphism from F to G is a function h : X Y such that g G (g h F ). We denote Mor(F, G) the set of the morphisms from F to G. F = Mor(F, R), where R = (R, Bic(R)).

7 Spanier s quasi-topological space is a structure (X, Q(K, X ) K chtop ), where chtop is the category of compact Hausdorff spaces and for every K, K, K 1, K 2 chtop the set of functions Q(K, X ) F(K, X ) satisfies the following conditions: (QT 1 ) x X x Q(K, X ). (QT 2 ) f Q(K, X ) g C(K, K) f g Q(K, X ). (QT 3 ) If g C(K, K) is a surjection, then f Q(K, X ) f g Q(K, X ). (QT 4 ) If K is the disjoint union of K 1, K 2 chtop, then f Q(K, X ) f K1 Q(K 1, X ) f K2 Q(K 2, X ). A mapping h : X Y, called a quasi-continuous function if K chtop f Q(K,X ) (h f Q(K, Y ))

8 Convergence as a primitive notion (Fréchet-Urysohn) A limit space is a pair (X, lim), where X is an inhabited set, and lim X X N is a relation such that: (i) If x X and (x) denotes the constant sequence x, then lim(x, (x)). (ii) α S (lim(x, x n) lim(x, x α(n) )). (iii) (Urysohn s axiom) If x X and x n X N, then α S β S (lim(x, x α(β(n)) )) lim(x, x n). Pattern: (i) Inclusion of the appropriate objects determined by the elements of the codomain of the functions studied (the elements of X in the case of a quasi-topological space, the elements of R in the case of Bishop spaces), (ii) Inclusion of the closure under composition with an already given set of functions, and (iii) Closure under some appropriate notion of approximation.

9 The least topology F(F 0 ) generated by a given subbase F 0 F(X ) f 0 F 0 f 0 F(F 0 ) f F(F 0 ), φ Bic(R) φ f F(F 0 ) a R a F(F 0 ) f, g F(F 0 ) f + g F(F 0 ), (g F(F 0 ), U(g, f, ɛ)) ɛ>0, f F(F 0 ) g 1 F(F 0 ) U(g 1, f, 1 2 ), g 2 F(F 0 ) U(g 2, f, ), g 3 F(F 0 ) U(g 3, f, ),... f0 F 0 (P(f 0 )) a R (P(a)) f F(F 0 ) f,g F(F0 )(P(f ) P(g) P(f + g)) f F(F0 ) φ Bic(R) (P(f ) P(φ f )) f F(F0 )( ɛ>0 g F(F0 )(P(g) U(g, f, ɛ)) P(f )) f F(F0 )(P(f )). If G 0 = (Y, F(G 0 )), then h : X Y Mor(F, G 0 ) if and only if g0 G 0 (g 0 h F ).

10 Product F G = (X Y, F G) of Bishop spaces F G := F({f π 1 f F } {g π 2 g G}) = {f π 1 f F } {g π 2 g G} = (F π 1 ) (G π 2 ). F i := F( π i f F i }) i I i I{f = i I{f π i f F i } = i I(F i π i ). F(F 0 ) F(G 0 ) = F({f 0 π 1 f 0 F 0 } {g 0 π 2 g 0 G 0 }). Bic(R) = F(id R ) Bic(R) Bic(R) = F({id R π 1 } {id R π 2 }) = F(id R π 1, id R π 2 ) = F(π 1, π 2 ).

11 Exponential F G = (X Y, F G), is defined by F G := F({e x,g x X, g G}), e x,g : Mor(F, G) R e x,g (h) = g(h(x)), for each h Mor(F, G). If G = (Y, F(G 0 )), then F G := F(E 0 ), E 0 := {e x,g0 x X, g 0 G 0 }, e x,g0 (h) = g 0 (h(x)). lim (h, hn) x X g G (e x,g (h n) e x,g (h)) F G x X g G (g(h n(x)) g(h(x))) x X (lim G (h(x), h n(x))).

12 The Bishop morphism as a generalization of uniform continuity 1. X, Y compact metric spaces, then a. h : X Y Mor(U(X ), U(Y )) h is uniformly continuous. b. C u(x ) C u(y ) = C u(x Y ), with the product metric. 2. (UCT) Mor(R [a,b], R) = C u([a, b]), where F A = F({f A f F }). 3. (FUCT) X is a compact metric space, Y is a metric space and G = (Y, F(U 0 (Y )), then h : X Y Mor(U(X ), G), then h is uniformly continuous. (It is proved by Bridges in BISH + AS.) 4. n N πn = Cu(2N ). 5. Mor(C, (N, F(N))) = C u(2 N, N).

13 The Cantor and the Baire space as Bishop spaces F(N) = Bic(N) = F(id N ) = Bic(R) N, F(2) = Bic(2) = F(id 2 ) = Bic(R) 2 C = (2 N, Bic(2) N ), N = (N N, Bic(N) N ), Bic(2) N = F({id 2 π n n N}) = F({π n n N}) = n N π n, Bic(N) N = F({id N ϖ n n N}) = F({ϖ n n N}) = n N ϖ n, ρ(α, β) := inf{2 n α(n) = β(n)}

14 Properties of the Cantor and the Baire space 1. α ρ β α n N πn β 2. α ρ β α n N ϖn β 3. ( n N ϖn) 2 N = n N πn 4. There is a retraction r of N N onto 2 N 5. C is embedded in N i.e., φ n N πn φ n N ϖn (φ 2 N = φ) 6. The finite or countably infinite products of the Bishop spaces N and C are Bishop-isomorphic to N and C, respectively 7. C is isomorphic to the Cantor set space C = (Ca, Bic[0, 1] Ca = F(id [0,1] Ca ) = F(id Ca )) These results do not depend on the metric structure of the Cantor and the Baire space.

15 The following φ : 2 N R n N π n Proof. { 1, if α(0) 0 φ(α) = 0, ow If we fix some ɛ (0, 1), we consider any real σ such that 0 < σ ɛ. In this case 1 ɛ we get that σ 1 = σ = σ 1 + σ ɛ. We also have that We define the function Next we show that 1 n 1 ( 1 + σ n n + σ < 1). g := π 0 π 0 + σ π n. n N α(0) U(g, φ, ɛ) := α 2 N ( g(α) φ(α) = φ(α) ɛ). α(0) + σ If α(0) = 0, then φ(α) = g(α) = 0, and we are done. If α(0) 0 α(0) = n 1, α(0) α(0)+σ φ(α) = n n+σ 1 = 1 n n+σ σ = σ 1+σ ɛ.

16 The following 2 N R functions are in n N π n { 1, if α(i) = β(i) θ α,i (β) = 0, ow { 2 i, if α(i) = β(i) η α,i (β) = 3, ow { 2 i, if α(i) = β(i) σ α,i (β) = 2 m, α(m) = β(m) and α(m) β(m) σ α,i = i η α,i j=1

17 {ρ α α 2 N } n N π n Proof. We fix some ɛ > 0 and let n 0 N such that 2 n 0 < ɛ. We show that U(σ α,n0, ρ α, ɛ) := β 2 N ( σ α,n0 (β) ρ α(β) ɛ). If α(n 0 ) = β(n 0 ), then ρ α(β) 2 n 0, and σ α,n0 (β) ρ α(β) = 2 n 0 ρ α(β) = 2 n 0 ρ α(β) 2 n 0 < ɛ. If α(n 0 ) β(n 0 ), then ρ α(β) = σ α,n0 (β) and we get that σ α,n0 (β) ρ α(β) = 0 ɛ. Since σ α,n0 n N πn, and ɛ > 0 is arbitrarily chosen, we get that U( n N πn, ρα), therefore by the condition BS 4 we conclude that ρ α n N πn By the Stone-Weierstrass theorem for the compact metric space (2 N, ρ) with diameter 1, we get F({ρ α α 2 N }) = C u(2 N ) n N π n

18 C u (2 N ) n N π n Proof. First we show that {π n n N} C u(2 N ). If we fix some n N and some 0 < ɛ < 1, we define ω πn (ɛ) = 2 n. If α, β 2 N such that ρ(α, β) < 2 n ρ(α, β) 2 (n+1), then α(n + 1) = β(n + 1), hence α(n) = β(n) and π n(α) π n(β) = α(n) β(n) = 0 < ɛ. Since every function π n is bounded, by the lifting of uniform continuity we get that n N πn Cu(2N ).

19 2-compact Bishop spaces A Bishop space (X, F ) is called 2-compact, if there exists some set I and a surjection e : 2 I X such that e Mor(2 I, F). Theorem Suppose that X is an inhabited compact metric space with positive diameter, and U(X ) = (X, C u(x )). If e : 2 N X is uniformly continuous, then e Mor(C, U(X )). Moreover, U(X ) is 2-compact. Proof. By the lifting of morphisms we have that e Mor(C, U(X )) x0 X (d x0 e n N πn). Since the composition of uniformly continuous functions is a uniformly continuous function, we get that d x0 e C u(2 N ) n N πn, for every x 0 X. Since X is an inhabited compact metric space, there exists a uniformly continuous function e from 2 N onto X. Hence e Mor(C, U(X )), and U(X ) is 2-compact.

20 Properties of 2-compact Bishop spaces 1. If F = (X, F ) is a 2-compact Bishop space, G = (Y, G) is a Bishop space and h : X Y Mor(F, G) is onto Y, then G is 2-compact. 2. If Y is an inhabited finite set, then (Y, F(Y )) is 2-compact. 3. If for every n N the Bishop space F n = (X n, F n) is 2-compact, then the product n N Fn is 2-compact. 4. If F = (X, F ) is 2-compact, then F is pseudo-compact. 5. There are 2-compact space (X, F ) for which it is not possible to accept constructively that f (X ) has a supremum, for some f F (generality of the index set I in the definition of 2-compactness).

21 Relevant literature E. Bishop: Foundations of Constructive Analysis, McGraw-Hill, D. S. Bridges: Reflections on function spaces, Annals of Pure and Applied Logic 163, 2012, H. Ishihara: Relating Bishop s function spaces to neighborhood spaces, Annals of Pure and Applied Logic 164, 2013, J. Myhill: Constructive Set Theory, J. Symbolic Logic 40, 1975, I. Petrakis: Completely Regular Bishop Spaces, in A. Beckmann, V. Mitrana and M. Soskova (Eds.): Evolving Computability, CiE 2015, LNCS 9136, 11 pages, DOI: / I. Petrakis: Constructive Topology of Bishop Spaces, PhD Thesis, LMU München, 2015.