Relative formal topology: the binary positivity predicate comes first

2011 ◽  
Vol 22 (1) ◽  
pp. 69-102 ◽  
Author(s):  
SILVIO VALENTINI
Keyword(s):  
Topological Space ◽  
Central Result ◽  
Definition Of ◽  
Cover Relation ◽  
Formal Topology

In this paper we introduce relative formal topology so that we can deal constructively with the closed subsets of a topological space as well as with the open ones. In fact, within standard formal topology, the cover relation allows the definition of the formal opens, which are supposed to act as the formal counterparts of the open subsets, and within balanced formal topology, the binary positivity predicate allows the definition of the formal closed subsets, which are supposed to act as the formal counterparts of the closed subsets. However, these approaches are only fully satisfactory (according to the criterion introduced by the author in Valentini (2005)) if we can provide an adequate formalisation of the cover relation and the positivity predicate. But current formalisations fail in this respect since some intuitionistically valid relations between the open and closed subsets of a concrete topological space cannot be expressed. Our central result is that we can solve this problem through a generalisation of the standard cover relation together with a binary positivity predicate satisfying the positivity axiom.

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 Objets compacts dans les topos

1986 ◽  
Vol 40 (2) ◽  
pp. 203-217 ◽  
Author(s):  
Eduardo J. Dubuc ◽  
Jacques Penon
Keyword(s):  
Differential Geometry ◽  
Topological Space ◽  
Compact Manifolds ◽  
Topological Spaces ◽  
Internal Logic ◽  
Definition Of

AbstractIt is well known that compact topological spaces are those space K for which given any point x0 in any topological space X, and a neighborhood H of the fibre -1 {x0} KXX, then there exists a neighborhood U of x0 such that -1UH. If now is an object in an arbitrary topos, in the internal logic of the topos this property means that, for any A in and B in K, we have (-1AB)=AB. We introduce this formula as a definition of compactness for objects in an arbitrary topos. Then we prove that in the gross topoi of algebraic, analytic, and differential geometry, this property characterizes exactly the complete varieties, the compact (analytic) spaces, and the compact manifolds, respectively.

scholarly journals Functional Space C(ω), C0(ω)

2012 ◽  
Vol 20 (1) ◽  
pp. 15-22
Author(s):  
Katuhiko Kanazashi ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Function Space ◽  
Normed Space ◽  
Functional Space ◽  
Continuous Functions ◽  
Bounded Support ◽  
Compact Topological Space ◽  
Definition Of ◽  
Complex Valued

Functional Space C(ω), C0(ω) In this article, first we give a definition of a functional space which is constructed from all complex-valued continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all complex-valued continuous functions with bounded support. We also prove that this function space is a complex normed space.

scholarly journals Relative Smooth Topological Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-5
Author(s):  
B. Ghazanfari
Keyword(s):  
Topological Space ◽  
Topological Spaces ◽  
General Properties ◽  
Fuzzy Topological Space ◽  
Definition Of

In 1992, Ramadan introduced the concept of a smooth topological space and relativeness between smooth topological space and fuzzy topological space in Chang's (1968) view points. In this paper we give a new definition of smooth topological space. This definition can be considered as a generalization of the smooth topological space which was given by Ramadan. Some general properties such as relative smooth continuity and relative smooth compactness are studied.

scholarly journals Banach Algebra of Continuous Functionals and the Space of Real-Valued Continuous Functionals with Bounded Support

2010 ◽  
Vol 18 (1) ◽  
pp. 11-16 ◽  
Author(s):  
Katuhiko Kanazashi ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Topological Space ◽  
Banach Algebra ◽  
Function Space ◽  
Functional Space ◽  
Continuous Functions ◽  
Bounded Support ◽  
Compact Topological Space ◽  
Real Normed Space ◽  
Definition Of ◽  
Continuous Functionals

Banach Algebra of Continuous Functionals and the Space of Real-Valued Continuous Functionals with Bounded Support In this article, we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space. We prove that this functional space is a Banach algebra. Next, we give a definition of a function space which is constructed from all real-valued continuous functions with bounded support. We prove that this function space is a real normed space.

scholarly journals Functional Space Consisted by Continuous Functions on Topological Space

2021 ◽  
Vol 29 (1) ◽  
pp. 49-62
Author(s):  
Hiroshi Yamazaki ◽  
Keiichi Miyajima ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Function Space ◽  
Normed Space ◽  
Functional Space ◽  
Continuous Functions ◽  
Bounded Support ◽  
Compact Topological Space ◽  
Definition Of ◽  
System 1

Summary In this article, using the Mizar system [1], [2], first we give a definition of a functional space which is constructed from all continuous functions defined on a compact topological space [5]. We prove that this functional space is a Banach space [3]. Next, we give a definition of a function space which is constructed from all continuous functions with bounded support. We also prove that this function space is a normed space.

Beth’s theorem and Craig’s theorem

2018 ◽  
Author(s):  
Zeno Swijtink
Keyword(s):  
Sufficient Conditions ◽  
Predicate Symbol ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Infinite Length ◽  
First Order ◽  
Explicit Definition ◽  
Central Result ◽  
Definition Of

Beth’s theorem is a central result about definability of non-logical symbols in classical first-order theories. It states that a symbol P is implicitly defined by a theory T if and only if an explicit definition of P in terms of some other expressions of the theory T can be deduced from the theory T. Intuitively, the symbol P is implicitly defined by T if, given the extension of these other symbols, T fixes the extension of the symbol P uniquely. In a precise statement of Beth’s theorem this will be replaced by a condition on the models of T. An explicit definition of a predicate symbol states necessary and sufficient conditions: for example, if P is a one-place predicate symbol, an explicit definition is a sentence of the form (x) (Px ≡φ(x)), where φ(x) is a formula with free variable x in which P does not occur. Thus, Beth’s theorem says something about the expressive power of first-order logic: there is a balance between the syntax (the deducibility of an explicit definition) and the semantics (across models of T the extension of P is uniquely determined by the extension of other symbols). Beth’s definability theorem follows immediately from Craig’s interpolation theorem. For first-order logic with identity, Craig’s theorem says that if φ is deducible from ψ, there is an interpolant θ, a sentence whose non-logical symbols are common to φ and ψ, such that θ is deducible from ψ, while φ is deducible from θ. Craig’s theorem and Beth’s theorem also hold for a number of non-classical logics, such as intuitionistic first-order logic and classical second-order logic, but fail for other logics, such as logics with expressions of infinite length.

Adic spaces II

2020 ◽  
pp. 17-26
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Topological Space ◽  
Definition Of

This chapter defines adic spaces. A scheme is a ringed space which locally looks like the spectrum of a ring. An adic space will be something similar. The chapter then identifies the adic version of “locally ringed space.” Briefly, it is a topologically ringed topological space equipped with valuations. The chapter also reflects on the role of A+ in the definition of adic spaces. The subring A+ in a Huber pair may seem unnecessary at first: why not just consider all continuous valuations on A? Specifying A+ keeps track of which inequalities have been enforced among the continuous valuations. Finally, the chapter differentiates between sheafy and non-sheafy Huber pairs.

The Hahn-Banach theorem in type theory

1998 ◽  
Author(s):  
Jan Cederquist ◽  
Thierry Coquand
Keyword(s):  
Linear Functional ◽  
Constructive Proof ◽  
Normed Vector Space ◽  
Deductive Systems ◽  
Banach Theorem ◽  
Cover Relation ◽  
The Right ◽  
Proof By Induction ◽  
Normed Vector ◽  
Formal Topology

We present the basic concepts and definitions needed in a pointfree approach to functional analysis via formal topology. Our main results are the constructive proofs of localic formulations of the Alaoglu and Helly-Hahn-Banach theorems. Earlier pointfree formulations of the Hahn-Banach theorem, in a topos-theoretic setting, were presented by Mulvey and Pelletier (1987, 1991) and by Vermeulen (1986). A constructive proof based on points was given by Bishop (1967). In the formulation of his proof, the norm of the linear functional is preserved to an arbitrary degree by the extension and a counterexample shows that the norm, in general, is not preserved exactly. As usual in pointfree topology, our guideline is to define the objects under analysis as formal points of a suitable formal space. After this has been accomplished for the reals, we consider the formal topology ℒ(A) obtained as follows. To the formal space of mappings from a normed vector space A to the reals, we add the linearity and norm conditions in the form of covering axioms. The linear functional of norm ≤1 from A to the reals then correspond to the formal points of this formal topology. Given a subspace M of A, the classical Helly-Hahn-Banach theorem states that the restriction mapping from the linear functionals on A of norm ≤1 to those on M is surjective. In terms of covers, conceived as deductive systems, it becomes a conservativity statement (cf. Mulvey and Pelletier 1991): whenever a is an element and U is a subset of the base of the formal space ℒ(M) and we have a derivation in ℒ(A) of a ⊲ U, then we can find a derivation in ℒ(M) with the same conclusion. With this formulation it is quite natural to look for a proof by induction on covers. Moreover, as already pointed out by Mulvey and Pelletier (1991), it is possible to simplify the problem greatly, since it is enough to prove it for coherent spaces of which ℒ(A) and ℒ(M) are retracts. Then, in a derivation of a cover, we can assume that only finite subsets occur on the right-hand side of the cover relation.

scholarly journals Relative Zariski Open Objects

2012 ◽  
Vol 10 (1) ◽  
pp. 9-39 ◽  
Author(s):  
Florian Marty
Keyword(s):  
Topological Space ◽  
Monoidal Category ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Scheme Theory ◽  
Usual Case ◽  
Definition Of ◽  
Affine Scheme

AbstractIn [TV], Bertrand Toën and Michel Vaquié define a scheme theory for a closed monoidal category (,⊗, 1) One of the key ingredients of this theory is the definition of a Zariski topology on the category of commutative monoidal objects in . The purpose of this article is to prove that under some hypotheses, Zariski open subobjects of affine schemes can be classified almost as in the usual case of rings (ℤ-mod,⊗,ℤ). The main result states that for any commutative monoidal object A in , the locale of Zariski open subobjects of the affine scheme Spec(A) is associated to a topological space whose points are prime ideals of A and whose open subsets are defined by the same formula as in rings. As a consequence, we can compare the notions of scheme over in [D] and in [TV].

Sign in / Sign up

Export Citation Format

Share Document