## Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability

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Venue: | Mathematical Logic Quarterly |

Citations: | 6 - 1 self |

### BibTeX

@ARTICLE{Bauer_twoconstructive,

author = {Andrej Bauer and Alex Simpson},

title = {Two constructive embedding-extension theorems with applications to continuity principles and to Banach-Mazur computability},

journal = {Mathematical Logic Quarterly},

year = {},

volume = {50},

pages = {351--369}

}

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### Abstract

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an analogous property for Baire space relative to any inhabited locally non-compact CSM. Both results rely on having careful constructive formulations of the concepts involved. As a first application, we derive new relationships between “continuity principles ” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to R are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non-compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1]. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space X, there exists a Banach-Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers.

### Citations

408 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...onstructive analysis, as they appear to provide general tools for extending properties of the Cantor and Baire spaces to other spaces. 2 Two constructive embedding-extension theorems Following Bishop =-=[4, 5]-=-, we do mathematics using intuitionistic logic, and we assume the principle of countable choice AC0, namely choice for statements of the form ∀ n ∈ N . ∃ x ∈ X . ϕ. We shall not need dependent choice.... |

264 | Foundations of constructive mathematics - Beeson - 1980 |

80 |
The effective topos
- Hyland
- 1982
(Show Context)
Citation Context ...domain pair in the sense of Section 3.2. Also, the function f ◦ ι : Bβ → Z is BM-computable hence, as observed above, computable. Thus f ◦ ι is a morphism in Eff . As CPpt(N + , N) holds in Eff , see =-=[11]-=-, CPseq(N + , N) holds as well by Proposition 4.3. Since Bβ as a subspace of Z N is a CSM by (7), it follows from Proposition 4.4 that f ◦ ι is sequentially continuous in the internal logic of Eff . T... |

45 |
Continuity and Effectiveness in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ... internal logic of Mulry’s topos is awkward to use; for example, it is necessary to work with a non-standard object of natural numbers, for which only restricted forms of induction are available, see =-=[19]-=-. Indeed, we do not know whether the proof of Theorem 2.4 goes through in this setting. Acknowledgement. We thank Peter Hertling, Peter Lietz, and Dag Normann for helpful comments and suggestions. 20s... |

41 | The Realizability Approach to Computable Analysis and Topology
- Bauer
- 2000
(Show Context)
Citation Context ...alizability”, in which the failure of the continuity principle for any inhabited locally non-compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C=-=[0, 1]-=-. As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion-theoretic setting, then, for any such space ... |

38 |
Constructive analysis, volume 279 of Grundlehren der Mathematischen Wissenschaften
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...nces of the corresponding results for Baire space, and we show that the results of [7, 10] both follow from applications of our method. Working within the context of Bishop’s Constructive Mathematics =-=[5]-=-, we identify two properties of complete separable metric spaces (CSMs), namely being without isolated points and local non-compactness. Despite the negative terminology, as befits the constructive se... |

37 |
The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions
- Kleene, Vesley
- 1965
(Show Context)
Citation Context ...N , N) and AC1,0 are valid. So the full power of Proposition 4.1.4 is available. This situation is mimicked within the internal logic of the realizability topos RT(K2) over Kleene’s second algebra K2 =-=[13, 1]-=-. Example 4.10 In Markov’s Recursive Mathematics, CPpt(Z N , R) is valid (although AC1,0 fails), and hence Proposition 4.1.4 is again available. This situation is mimicked within the internal logic of... |

24 |
The Theory of Triposes
- Pitts
- 1981
(Show Context)
Citation Context ...se continuous functions from [−1, 1]. In RT(D), it holds that C[−1, 1] = Cu[−1, 1], using AC1,0 and the uniform continuity of functions from 2 N to N.) Example 4.12 In the extensional effective topos =-=[18, 22]-=-, by construction, choice holds for all “finite types”; in particular, AC1,0 and AC2,0 hold. Peter Lietz [14, Chapter II] shows that CPseq(N + , N) holds, but that CPpt(2 N , N) fails. By Proposition ... |

21 |
On a topological topos
- Johnstone
- 1979
(Show Context)
Citation Context ...uch a meta-theorem would be to develop a constructive analogue of Johnstone’s “topological topos” T of sheaves for the canonical Grothendieck topology on the monoid of continuous endomorphisms on N + =-=[12]-=-. Then the relativization of the construction of h to T would a fortiori result in a sequentially continuous function being produced. It would be interesting to see this worked out in detail. Remark 3... |

16 |
The eective topos
- Hyland
- 1982
(Show Context)
Citation Context ...omain pair in the sense of Section 3.2. Also, the function f # # : B # # Z is BM-computable hence, as observed above, computable. Thus f # # is a morphism in E# . As CP pt (N + , N) holds in E# , see =-=[11]-=-, CP seq (N + , N) holds as well by Proposition 4.3. Since B # as a subspace of Z N is a CSM by (7), it follows from Proposition 4.4 that f # # is sequentially continuous in the internal logic of E# .... |

16 |
Continuity and E#ectivity in Topoi
- Rosolini
- 1986
(Show Context)
Citation Context ... internal logic of Mulry's topos is awkward to use; for example, it is necessary to work with a non-standard object of natural numbers, for which only restricted forms of induction are available, see =-=[19]-=-. Indeed, we do not know whether the proof of Theorem 2.4 goes through in this setting. Acknowledgement. We thank Peter Hertling, Peter Lietz, and Dag Normann for helpful comments and suggestions. 20 ... |

14 | From Constructive Mathematics to Computable Analysis via the Realizability Interpretation - Lietz - 2004 |

14 |
Generalized Banach-Mazur functionals in the topos of recursive sets
- Mulry
- 1982
(Show Context)
Citation Context ...sential to our proof because the BM-computable functions do not live inside Eff . An interesting alternative would be to instead apply Theorem 2.4 directly in the context of Mulry’s “recursive topos” =-=[16]-=-, in which the morphisms (between certain objects) are exactly the BM-computable functions. Such an approach may be possible, but it is non-trivial because the internal logic of Mulry’s topos is awkwa... |

12 | Continuous functionals of dependent types and equilogical spaces
- Bauer, Birkedal
- 2000
(Show Context)
Citation Context ...s of uniform continuity on Cantor space 2 N . (In fact a stronger continuity principle holds: all functions from 2 N to N are uniformly continuous.) Also choice holds between arbitrary “finite types” =-=[2, 1]-=-), in particular AC1,0 and AC2,0 hold. Thus, by Proposition 4.2, CPpt(X, Y ) holds, for every locally CTB CSM X and metric space Y . On the other hand, by Corollary 4.8, CPpt(X, R) fails, for any inha... |

9 |
Extensional realizability
- Oosten
- 1993
(Show Context)
Citation Context ...se continuous functions from [−1, 1]. In RT(D), it holds that C[−1, 1] = Cu[−1, 1], using AC1,0 and the uniform continuity of functions from 2 N to N.) Example 4.12 In the extensional effective topos =-=[18, 22]-=-, by construction, choice holds for all “finite types”; in particular, AC1,0 and AC2,0 hold. Peter Lietz [14, Chapter II] shows that CPseq(N + , N) holds, but that CPpt(2 N , N) fails. By Proposition ... |

4 |
4-quantifier completeness: A Banach-Mazur functional not uniformly partial recursive
- Friedberg
- 1958
(Show Context)
Citation Context ... it is not easy to find counterexamples. A first (and sophisticated) such example was produced by Friedberg, who showed that there exists a BM-computable function from N N to N that is not computable =-=[8]-=-. A general discussion of the relationship between the two notions is contained in Hertling’s recent paper [10], where, as the main result, a BM-computable but non-computable function from Rc to Rc is... |

4 |
A Banach-Mazur computable but not Markov computable function on the computable real numbers
- Hertling
(Show Context)
Citation Context ...ble). Recently, Hertling answered a longstanding open question by proving that, similarly, there exists a Banach-Mazur-computable function on the computable real numbers that is not Markov computable =-=[10]-=-. In neither example above [7, 10] is the pathological behaviour in the analytic world derived from the analogous result for Baire space. Instead, direct proofs are given, borrowing ideas from the kno... |

4 |
Computable Analysis, volume 33. Razprawy Matematyczne
- Mazur
- 1963
(Show Context)
Citation Context ...s, by cartesian closure, m ↦→ λn. f(n, xm) : N → Y N is computable, i.e. m ↦→ f(xm) is computable, as required. We omit the similar proof of statement 2. (Anyway, the result is not used below.) Mazur =-=[15]-=- proved that every BM-computable functions from Rc to Rc enjoys the property that, for any computable Cauchy sequence (xn)n with limit x∞, it holds that (f(xn))n, considered as a sequence of ordinary ... |

4 |
Comparing hierarchies of total functionals. Unpublished manuscript
- Normann
- 2003
(Show Context)
Citation Context ...this approach to extending functionals has been worked out in a classical setting by Normann, who has embedded the entire continuous type hierarchy over N in the continuous type hierarchy over R, see =-=[17]-=-. Remark 3.18 It should be possible to avoid the technical proof that h is sequentially continuous, by proving a meta-theorem guaranteeing that, because h is defined constructively from functions that... |

3 |
domain realizability, not all functionals on C[−1, 1] are continuous
- In
- 1965
(Show Context)
Citation Context ...because it conflicts with choice principles valid in the models. Recently, Escardó and Streicher showed that similarly the internal statement “all functions from C[0, 1] to R are continuous” is false =-=[7]-=-. Once again, externally, all morphisms from C[0, 1] to R are continuous. ∗Faculty of Mathematics and Physics, University of Ljubljana, Slovenia. E-mail: Andrej.Bauer@andrej.com. Research supported by... |

2 |
Banach-Mazur computable functions on metric spaces
- Hertling
(Show Context)
Citation Context ...putable, is obviously not provable using only constructive principles consistent with Church’s Thesis, CT0 [20, 4.3]. 15s5.1 Numbered sets, Markov computability and computable metric spaces Following =-=[9, 10]-=-, we introduce the notion of Markov and Banach-Mazur computability in the setting of computable metric spaces presented as numbered sets. A numbered set, also known as modest set, is a structure X = (... |