## Resolution of the uniform lower bound problem in constructive analysis (2007)

Citations: | 3 - 3 self |

### BibTeX

@MISC{Palmgren07resolutionof,

author = {Erik Palmgren and Erik Palmgren},

title = {Resolution of the uniform lower bound problem in constructive analysis},

year = {2007}

}

### OpenURL

### Abstract

constructive analysis

### Citations

461 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...2 Keywords: Locales, formal topologies, locally compact metric spaces It is a well-known problem in constructive analysis to find a positive uniform lower bound of a uniformly continuous function f : =-=[0,1]-=- → (0,∞). Unlike in classical analysis this is not always possible, as shown by a computable counterexample due to Julian and Richman [7]; see also [4]. For the case of point-wise continuous functions... |

62 |
F.:Varieties of constructive mathematics
- Bridges, Richman
- 1987
(Show Context)
Citation Context ...r bound of a uniformly continuous function f : [0,1] → (0,∞). Unlike in classical analysis this is not always possible, as shown by a computable counterexample due to Julian and Richman [7]; see also =-=[4]-=-. For the case of point-wise continuous functions such an example was already constructed in a celebrated paper by Specker [10]. The uniform lower bound problem is also equivalent to a version of Brou... |

42 |
Nicht konstruktiv beweisbare Sätze der Analysis
- Specker
- 1949
(Show Context)
Citation Context ...wn by a computable counterexample due to Julian and Richman [7]; see also [4]. For the case of point-wise continuous functions such an example was already constructed in a celebrated paper by Specker =-=[10]-=-. The uniform lower bound problem is also equivalent to a version of Brouwer’s Fan Theorem [7, Corollary 2.6]; see also Bridges [2]. In this paper we aim to show that the problem is resolved in the fr... |

40 | Constructive Functional Analysis - Bridges - 1979 |

13 | Localic completion of generalized metric spaces II, Journal of Logic and Analysis 1:11(2009), 1 – 49; doi:10.4115/jla.2009.1.11
- Vickers
- 2010
(Show Context)
Citation Context ... ⊆ X ×Y , which is an abstraction and a constructive version of the relation f [U] ⊆ V for a continuous map. We refer to one of the papers in [5, 9] for precise definitions and basic results. Vickers =-=[11, 12]-=- gives a construction M (X) of a formal topology from a metric space X so that jX : X → Pt(M (X)) is a metric completion of X. Consequently jX is a metric isomorphism, in case X is already complete. W... |

10 | Continuity on the real line and in formal spaces
- Palmgren
- 2005
(Show Context)
Citation Context ...procal function with a uniformly continuous function R → (0,∞) need not produce a uniformly continuous function, is avoided in the point-free approach. This is because it admits fewer functions. (See =-=[8]-=- for the point-free reciprocal.) In intuitionistic analysis this not a problem since the Fan Theorem will provide a positive uniform lower bound and, even directly, produce a uniformly continuous func... |

9 |
Localic Completion of Generalized Metric Spaces I, Theory and
- Vickers
- 2005
(Show Context)
Citation Context ... ⊆ X ×Y , which is an abstraction and a constructive version of the relation f [U] ⊆ V for a continuous map. We refer to one of the papers in [5, 9] for precise definitions and basic results. Vickers =-=[11, 12]-=- gives a construction M (X) of a formal topology from a metric space X so that jX : X → Pt(M (X)) is a metric completion of X. Consequently jX is a metric isomorphism, in case X is already complete. W... |

8 | A constructive and functorial embedding of locally compact metric spaces
- Palmgren
(Show Context)
Citation Context ...us morphism of formal topologies F : X → Y is a relation R ⊆ X ×Y , which is an abstraction and a constructive version of the relation f [U] ⊆ V for a continuous map. We refer to one of the papers in =-=[5, 9]-=- for precise definitions and basic results. Vickers [11, 12] gives a construction M (X) of a formal topology from a metric space X so that jX : X → Pt(M (X)) is a metric completion of X. Consequently ... |

6 |
A uniformly continuous function on [0, 1] that is everywhere different from its infimum
- Julian, Richman
- 1984
(Show Context)
Citation Context ...e uniform lower bound of a uniformly continuous function f : [0,1] → (0,∞). Unlike in classical analysis this is not always possible, as shown by a computable counterexample due to Julian and Richman =-=[7]-=-; see also [4]. For the case of point-wise continuous functions such an example was already constructed in a celebrated paper by Specker [10]. The uniform lower bound problem is also equivalent to a v... |

1 |
On continuous mappings between locally compact metric spaces
- Bridges
(Show Context)
Citation Context ...an example was already constructed in a celebrated paper by Specker [10]. The uniform lower bound problem is also equivalent to a version of Brouwer’s Fan Theorem [7, Corollary 2.6]; see also Bridges =-=[2]-=-. In this paper we aim to show that the problem is resolved in the framework of point-free topology or locale theory. In this framework the notion of positivity has a stronger meaning. This provides a... |

1 |
eds.) Special issue with papers from the second workshop on formal topology
- Coquand, Sambin
- 2006
(Show Context)
Citation Context ...us morphism of formal topologies F : X → Y is a relation R ⊆ X ×Y , which is an abstraction and a constructive version of the relation f [U] ⊆ V for a continuous map. We refer to one of the papers in =-=[5, 9]-=- for precise definitions and basic results. Vickers [11, 12] gives a construction M (X) of a formal topology from a metric space X so that jX : X → Pt(M (X)) is a metric completion of X. Consequently ... |