Formal Topology Research Articles

Generative design of space frames for additive manufacturing technology

A generative design methodology is presented that solves for minimum volume and compliance space-frame systems, with consideration of stress and buckling constraints. The solution space is explored using formal topology optimisation routines. A parameterisation method converts voxelised topology optimisation solutions into skeletonised connectivity representations. An inequality constrained gradient descent optimisation method optimises and defines cross-sectional geometry. This enables fast and automatic solution generation, providing designers with sets of high-performing problem solutions. Skeleton representations provide an inexpensive modelling tool for parallel analysis of physical problems difficult to model using topology optimisation. Geometry is represented using traditional engineering cross-sections with well understood behaviour. This improves certainty in the performance of solutions, simplifying certification processes. The generative design of a structural aerospace bracket for additive manufacture has been used as a case study within this research. A design of experiments produced 360 topology optimisation results, altering input variables and discretisation resolution to identify their effect on solution outcomes and the performance of parameterisation. The proposed method was found to robustly generate high-performing solutions utilising low-resolution topology optimisation. Additionally, 6 high-performing topologies were identified, providing designers with a set of solutions to select from. Limitations on the parameterisation process were identified, with topology optimisation solutions with volume fractions above 0.2 not parameterising successfully, and simulations with a resolution of 5 mm element size and below generating low performing skeletonised topologies.

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-L\"of-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-L\"of's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.

$\sigma$-locales in Formal Topology

A $\sigma$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $\sigma$-sublocale of a given $\sigma$-locale, thus providing a "$\sigma$-version" of a Boolean locale. Our development depends on the axiom of countable choice.

Preface to the special issue for The Fifth Workshop on Formal Topology

Preface to the special issue for The Fifth Workshop on Formal Topology

Automorphisms of the completion of relatively free Lie algebras

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

Embedding locales and formal topologies into positive topologies

A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a reflective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales.

Eliminating disjunctions by disjunction elimination

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation due to Scott.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which turn out provable in a wealth of mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum and Szpilrajn, as well as of the spatiality of coherent locales. Related work has been done before on individual instances, e.g. in locale theory, dynamical algebra, formal topology and proof analysis.

Formal Topology and Constructive Mathematics: the Gelfand and Stone-Yosida Representation Theorems

We present a constructive proof of the Stone-Yosida representation theorem for Riesz spaces motivated by considerations from formal topology. This theorem is used to derive a representation theorem for f-algebras. In turn, this theorem implies the Gelfand representation theorem for C*-algebras of operators on Hilbert spaces as formulated by Bishop and Bridges. Our proof is shorter, clearer, and we avoid the use of approximate eigenvalues.

ELIMINATING DISJUNCTIONS BY DISJUNCTION ELIMINATION

AbstractCompleteness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation given by Scott in 1974.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which in turn prove admissible in all known mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include the syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum, and Szpilrajn. Related work has been done before on individual instances, e.g., in locale theory, dynamical algebra, formal topology and proof analysis.

Point-free characterisation of Bishop compact metric spaces

We give a point-free characterisation of Bishop compact metric spaces in terms of formal topology. We show that the notion of overt compact enumerably completely regular formal topology is a point-free counterpart of that of Bishop compact metric space. Specifically, a formal topology is isomorphic to an overt compact enumerably completely regular formal topology if and only if it is isomorphic to the image of a compact metric space under the localic completion of metric spaces into formal topologies. The result is obtained in Bishop constructive mathematics with the axiom of Dependent Choice.

A point-free characterisation of Bishop locally compact metric spaces

We give a characterisation of Bishop locally compact metric spaces in terms of formal topology. We identify inhabited enumerably locally compact regular formal topologies as point-free counterpart of Bishop locally compact metric spaces. Specifically, we show that a formal topology is isomorphic to an inhabited enumerably locally compact regular formal topology if and only if it is isomorphic to the localic completion of a Bishop locally compact metric space. The result is obtained in Bishop constructive mathematics with the Dependent Choice.

Positivity relations on a locale

This paper analyses the notion of a positivity relation of Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad).

The Basic Zariski Topology

We present the Zariski spectrum as an inductively generated basic topology à la Martin-Löf and Sambin. Since we can thus get by without considering powers and radicals, this simplifies the presentation as a formal topology initiated by Sigstam. Our treatment includes closed subspaces and basic opens: that is, arbitrary quotients and singleton localisations. All the effective objects under consideration are introduced by means of inductive definitions. The notions of spatiality and reducibility are characterized for the class of Zariski formal topologies, and their nonconstructive content is pointed out: while spatiality implies classical logic, reducibility corresponds to a fragment of the Axiom of Choice in the form of Russell’s Multiplicative Axiom.

Generalized geometric theories and set-generated classes

We introduce infinitary propositional theories over a set and their models which are subsets of the set, and define a generalized geometric theory as an infinitary propositional theory of a special form. The main result is thatthe class of models of a generalized geometric theory is set-generated. Here, a class$\mathcal{X}$of subsets of a set is set-generated if there exists a subsetGof$\mathcal{X}$such that for each α ∈$\mathcal{X}$, and finitely enumerable subset τ of α there exists a subset β ∈Gsuch that τ ⊆ β ⊆ α. We show the main result in the constructive Zermelo–Fraenkel set theory (CZF) with an additional axiom, called the set generation axiom which is derivable inCZF, both from the relativized dependent choice scheme and from a regular extension axiom. We give some applications of the main result to algebra, topology and formal topology.

The axiom of multiple choice and models for constructive set theory

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory (hence acceptable from a constructive and generalized-predicative standpoint). In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.

A constructive notion of codimension

We give a point-free and constructively meaningful characterization of the notion of codimension for ideals of a commutative ring, which with classical logic and the axiom of choice is equivalent to the customary one. This characterization is based on notions from formal topology, and upon the elementary characterization of Krull dimension provided earlier by Coquand, Lombardi, and Roy. Among other things, we use this notion to prove Krullʼs principal ideal theorem in the context of constructive algebra.

Convergence in formal topology: a unifying notion

Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive genera- tion of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone's coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative co-semigroups over the category of basic covers (suplattices).

About Goodmanʼs Theorem

We present a proof of Goodmanʼs Theorem, which is a variation of the proof of Renaldel de Lavalette (1990) [9]. This proof uses in an essential way possibly divergent computations for proving a result which mentions systems involving only terminating computations. Our proof is carried out in a constructive metalanguage. This involves implicitly a covering relation over arbitrary posets in formal topology, which occurs in forcing in set theory in a classical framework, but can also be defined constructively.

Cantor theorem and friends, in logical form

We prove a generalization of the hyper-game theorem by using an abstract version of inductively generated formal topology. As applications we show proofs for Cantor theorem, uncountability of the set of functions from N to N and Gödel theorem which use no diagonal argument.

Non-deterministic inductive definitions

We study a new proof principle in the context of constructive Zermelo-Fraenkel set theory based on what we will call "non-deterministic inductive definitions". We give applications to formal topology as well as a predicative justification of this principle.