scholarly journals Convergence in formal topology: a unifying notion

2013 ◽  
pp. 1-45 ◽  
Author(s):  
Francesco Ciraulo ◽  
Maria Emilia Maietti ◽  
Giovanni Sambin
Keyword(s):  
Formal Topology

Minimal invariant spaces in formal topology

1997 ◽  
Vol 62 (3) ◽  
pp. 689-698 ◽  
Author(s):  
Thierry Coquand
Keyword(s):  
Direct Consequence ◽  
Arithmetical Progression ◽  
Standard Result ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Combinatorial Theorem ◽  
Minimal Property ◽  
Special Instance ◽  
Invariant Spaces ◽  
Formal Topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces 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 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.

scholarly journals Compactness in locales and in formal topology

2006 ◽  
Vol 137 (1-3) ◽  
pp. 413-438 ◽  
Author(s):  
Steven Vickers
Keyword(s):  
Formal Topology

Tychonoff's theorem in the framework of formal topologies

1997 ◽  
Vol 62 (4) ◽  
pp. 1315-1332 ◽  
Author(s):  
Sara Negri ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
Constructive Proof ◽  
Topological Product ◽  
Essential Difference ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Inductive Property ◽  
Formal Topology

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

scholarly journals Formalization of Formal Topology by Means of the Interactive Theorem Prover Matita

2011 ◽  
pp. 278-280
Author(s):  
Andrea Asperti ◽  
Maria Emilia Maietti ◽  
Claudio Sacerdoti Coen ◽  
Giovanni Sambin ◽  
Silvio Valentini
Keyword(s):  
Theorem Prover ◽  
Formal Topology
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