## Minimal Invariant Spaces in Formal Topology (1996)

Venue: | The Journal of Symbolic Logic |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Coquand96minimalinvariant,

author = {Thierry Coquand},

title = {Minimal Invariant Spaces in Formal Topology},

journal = {The Journal of Symbolic Logic},

year = {1996},

volume = {62},

pages = {689--698}

}

### OpenURL

### Abstract

this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f : X!X: We show that such minimal subspaces can be described as points of a suitable formal topology, and the "existence" of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topological proof [6, 8], but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements. 1 Construction of Minimal Invariant Subspace

### Citations

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Citation Context ... is consistent and positive. These are purely syntactical properties that can be shown in a relatively weak constructive metalanguage. This can be seen as an illustration of some remarks contained in =-=[1]-=-. Conclusion We have given a completely elementary and algebraic proposition that replaces in a given concrete application the existence of minimal invariant closed subset of a boolean space with a co... |

25 |
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(Show Context)
Citation Context ...show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is "similar in structure" to the topologi=-=cal proof [6, 8]-=-, but which uses a simple algebraic remark (proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/e... |

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Citation Context ...icate over B that satisfies these properties, then the closed subset that is the intersection of all clopen satisfying �� is a closed minimal invariant subset. 1.2 Space of minimal subspace Follow=-=ing [15], we can s-=-ee the 6 properties as describing "forcing" conditions on a point of a space. This space M can be seen as an infinitary propositional logic defined inductively by the properties 1. x ` g(x),... |

18 |
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(Show Context)
Citation Context ...next a covering relation on the set of clopen of the space X : x \Delta U = (8z)[[(8y 2 U)Z I (z:y)] ) Z I (z:x)]: This defines a formal space M I , following Sambin's definition of a formal topology =-=[16]-=-. Proposition 2: The relation x \Delta U satisfies ffl if x 2 U; then x \Delta U; ffl if x \Delta U; and u \Delta V for all u 2 U; then x \Delta V; ffl if x \Delta U and x \Delta V; then x \Delta U:V;... |

8 |
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Citation Context ...tion that the space X has a point. This can be expressed as follows: if a concrete statement (like the statement above) is valid in a "relativised" sense, namely interpreted in the 2 See for=-= instance [14]-=- for one example of this technique; as shown in [5], this method can be used even in cases where, even classically, the formal space fail to have any point. An example is the formal space of surjectiv... |

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(Show Context)
Citation Context ...spaces K of X such that f(K) ` K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work =-=[3]-=-, we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression). In this pap... |

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1 |
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Citation Context ...o a continuous map f on the space X of ultrafilters, and �� corresponds to an invariant non empty invariant subset. Any point of this subset defines a non principal ultrafilter. We refer to the pa=-=per [4]-=- for a discussion on the analysis of the notion of ultrafilters in formal topology. 3 A Reformulation of Hilbert's Program One important component of Hilbert's program [9] is the following justificati... |

1 |
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Citation Context ...ssed as follows: if a concrete statement (like the statement above) is valid in a "relativised" sense, namely interpreted in the 2 See for instance [14] for one example of this technique; as=-= shown in [5]-=-, this method can be used even in cases where, even classically, the formal space fail to have any point. An example is the formal space of surjective functions from natural numbers to a set X: This f... |

1 |
der Meiden Point-free Carrier Space Topology for Commutative Banach Algebras
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Citation Context ..., and in particular, avoids the use of Zorn's lemma. Such a remark about similarity of proofs appear already in an early appplication of point-free topology in avoiding the use of the axiom of choice =-=[2, 13] in the develop-=-ment of the theory of Banach algebras. In these works the notion of "topology without points" is used to give "a theory entirely parallel to Gelfand's, such that it is possible at every... |