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SEPARATION PROPERTIES IN CONSTRUCTIVE TOPOLOGY

Peter Aczel and Christopher Fox

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

This chapter studies separation properties in topology as done on the basis of the formal system CZF. Until the 1970s, there was only a limited focus on the general notion of a topological space in ... More

This chapter studies separation properties in topology as done on the basis of the formal system CZF. Until the 1970s, there was only a limited focus on the general notion of a topological space in constructive mathematics, with most attention being paid to metric space notions both in intuitionistic analysis and in Bishop-style constructive analysis. But in later years, because of the development of topos theory, the study of sheaf models, and work on point-free topology, including work on formal topology, the notions of general topology in constructive mathematics have received more attention. This chapter presents a fairly systematic survey of the main separation properties that topological spaces can have in constructive mathematics. These are perhaps the first kinds of properties to consider when moving from the study of metric spaces to the study of general topological spaces.Less

SPACES AS COMONOIDS

A. Bucalo and G. Rosolini

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

The constructive approach to topology has produced a wealth of new insights about spaces, almost reaching the point of making these more elementary than sets. In particular, the point of view taken ... More

The constructive approach to topology has produced a wealth of new insights about spaces, almost reaching the point of making these more elementary than sets. In particular, the point of view taken by the notion of the basic picture as developed in the context of formal topology brings forward the relevance of relations between sets as an elementary constituent for understanding topological spaces. The chapter employs the language and notions of category theory to put the concept of the basic pair under scrutiny. It gives a presentation of topological spaces, which exploits another monoidal structure on the category of complete sup-lattices compared with the one given by Galois connections as in the memoirs by Joyal and Tierney. In this modified context, this chapter characterizes the frames which are topologies.Less

Essentials of General Topology

Adel N. Boules

in Fundamentals of Mathematical Analysis

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of ... More

The first eight sections of this chapter constitute its core and are generally parallel to the leading sections of chapter 4. Most of the sections are brief and emphasize the nonmetric aspects of topology. Among the topics treated are normality, regularity, and second countability. The proof of Tychonoff’s theorem for finite products appears in section 8. The section on locally compact spaces is the transition between the core of the chapter and the more advanced sections on metrization, compactification, and the product of infinitely many spaces. The highlights include the one-point compactification, the Urysohn metrization theorem, and Tychonoff’s theorem. Little subsequent material is based on the last three sections. At various points in the book, it is explained how results stated for the metric case can be extended to topological spaces, especially locally compact Hausdorff spaces. Some such results are developed in the exercises.Less

CONTINUITY ON THE REAL LINE AND IN FORMAL SPACES

Erik Palmgren

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Brouwer introduced his axioms for intuitionism to regain central results on continuity. Special axioms were avoided instead in Bishop's development of constructive analysis. Bishop in fact modified ... More

Brouwer introduced his axioms for intuitionism to regain central results on continuity. Special axioms were avoided instead in Bishop's development of constructive analysis. Bishop in fact modified the definition of continuous function on the real numbers to mean uniformly continuous on each finite and closed interval. Though a successful move in the context of metric spaces, this seems to lead to difficulties when considering general spaces, in particular as the composition of two continuous functions needs not to be continuous. Though little emphasized, the continuous functions of the category of locales or formal spaces agree with Bishop's definition of continuous function on real numbers. Proving this within the framework of (Bishop) constructive mathematics is the purpose of the present chapter. The upshot is that for formal spaces, it is not necessary to adopt special axioms to obtain a good category.Less

Overview

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces ... More

This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.Less

Adic spaces II

Peter Scholze and Jared Weinstein

in Berkeley Lectures on p-adic Geometry: (AMS-207)

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 ... More

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.Less

Equivariant Cohomology of S² Under Rotation

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by ... More

This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by rotation. The method of the chapter only gives the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient MG fits into Cartan's mixing diagram. One can then apply Leray's spectral sequence of the fiber bundle MG → BG to compute the equivariant cohomology from the cohomology of M and the cohomology of the classifying space BG.Less

Topology

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. Product spaces, the ... More

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. Product spaces, the product topology, and the Tychonoff theorem are treated and also ideas of embedding, compactification, and metrization.Less

Symmetric Markovian Semigroups and Dirichlet Forms

Zhen-Qing Chen and Masatoshi Fukushima

in Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the ... More

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the correspondence of the above-mentioned notions to the semigroups of symmetric Markovian linear operators. Later on the chapter assumes that E is a Hausdorff topological space and considers the semigroups and Dirichlet forms generated by symmetric Markovian transition kernels on E. The chapter also considers quasi-regular Dirichlet forms and the quasi-homeomorphism of Dirichlet spaces. From here, the chapter shows that there is a nice Markov process called an m-tight special Borel standard process associated with every quasi-regular Dirichlet form.Less

Diamonds associated with adic spaces

Peter Scholze and Jared Weinstein

in Berkeley Lectures on p-adic Geometry: (AMS-207)

This chapter focuses on diamonds associated with adic spaces. The goal is to construct a functor which forgets the structure morphism to Spa Zp, but retains topological information. The chapter ... More

This chapter focuses on diamonds associated with adic spaces. The goal is to construct a functor which forgets the structure morphism to Spa Zp, but retains topological information. The chapter studies how much information is lost when applying this construction. The intuition is that only topological information is kept. A morphism of adic spaces is a universal homeomorphism if all pullbacks are homeomorphisms. As in the case of schemes, in characteristic 0 the map f is a universal homeomorphism if and only if it is a homeomorphism and induces isomorphisms on completed residue fields. In keeping with the intuition, universal homeomorphisms induce isomorphisms of diamonds. The chapter then considers the underlying topological space of diamonds, as well as the étale site of diamonds.Less

Entropy in dimension one

William P. Thurston

Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)

in Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions ... More

This chapter studies the topological entropy h of postcritically finite one-dimensional maps and, in particular, the relations between dynamics and arithmetics of eʰ, presenting some constructions for maps with given entropy and characterizing what values of entropy can occur for postcritically finite maps. In particular, the chapter proves: h is the topological entropy of a postcritically finite interval map if and only if h = log λ‎, where λ‎ ≥ 1 is a weak Perron number, i.e., it is an algebraic integer, and λ‎ ≥ ∣λ‎superscript Greek Small Letter Sigma∣ for every Galois conjugate λ‎superscript Greek Small Letter Sigma ∈ C. Unfortunately, the author of this chapter has died before completing this work, hence this chapter contains both the original manuscript as well as a number of notes which clarify many of the points mentioned therein.Less

Transcendental Deduction II: The B Edition Transcendental Deduction

Thomas C. Vinci

in Space, Geometry, and Kant's Transcendental Deduction of the Categories

Chapter 7 has two parts. Part I deals with Kant on apperception, a second-order cognitive power that reveals propositional structure in all unified representations, including propositional structure ... More

Chapter 7 has two parts. Part I deals with Kant on apperception, a second-order cognitive power that reveals propositional structure in all unified representations, including propositional structure in judgments of perception. Beginning in section 16, Kant constructs a sequence of arguments that concludes—in section 20—that an intuition in general “necessarily stands under the categories.” Part II has three main divisions. Following Allison, the first shows that intuitions in general and empirical intuitions are disjoint classes and that sections 22–27 show that empirical intuitions have the unity of the categories. The second division shows how this is carried out, arguing that it depends on Kant’s Nomic Prescriptivism. The third gives Kant’s argument for Nomic Prescriptivism, revealing its structure as analogous to the Second Geometrical Argument for Transcendental Idealism, a point first made by Ameriks, but focused on space’s topological unity rather than its geometry and employing the Table of Categories.Less

Homotopy Groups and CW Complexes

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built ... More

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.Less

The space of stably dominated types

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

This chapter introduces the space unit vector V of stably dominated types on a definable set V. It first endows unit vector V with a canonical structure of a (strict) pro-definable set before ... More

This chapter introduces the space unit vector V of stably dominated types on a definable set V. It first endows unit vector V with a canonical structure of a (strict) pro-definable set before providing some examples of stably dominated types. It then endows unit vector V with the structure of a definable topological space, and the properties of this definable topology are discussed. It also examines the canonical embedding of V in unit vector V as the set of simple points. An essential feature in the approach used in this chapter is the existence of a canonical extension for a definable function on V to unit vector V. This is considered in the next section where continuity criteria are given. The chapter concludes by describing basic notions of (generalized) paths and homotopies, along with good metrics, Zariski topology, and schematic distance.Less

v-sheaves associated with perfect and formal schemes

Peter Scholze and Jared Weinstein

in Berkeley Lectures on p-adic Geometry: (AMS-207)

This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also ... More

This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let X be any pre-adic space over Zp. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that k is algebraically closed.Less

An equivalence of categories

Ehud Hrushovski and François Loeser

in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable ... More

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable homotopy category of definable spaces over the o-minimal Γ‎. The chapter introduces three categories that can be viewed as ind-pro definable and admit natural functors to the category TOP of topological spaces with continuous maps. The discussion is often limited to the subcategory consisting of A-definable objects and morphisms. The morphisms are factored out by (strong) homotopy equivalence. The chapter presents the proof of the equivalence of categories before concluding with remarks on homotopies over imaginary base sets.Less

Frames of reference used in goal-directed arm movement

Sylvie Vanden Abeele, Marc Crommelinck, and André Roucoux

in Multisensory Control of Movement

This chapter discusses the frames of reference used in goal-directed arm movement. To generate a hand movement toward a visual stimulus, various sensory messages must be collected and selected by the ... More

This chapter discusses the frames of reference used in goal-directed arm movement. To generate a hand movement toward a visual stimulus, various sensory messages must be collected and selected by the brain in order to build up the appropriate motor commands that will steer the limb toward the target. It is usually postulated that both sensory information and motor program are referred to an internal representation of space, and interfacing them implies at least two important stages. First, the target's image projected onto the retinal map has to be relocated onto a motor (or visuomotor) map of space in which target position is specified in a body-centred system of coordinates. Second, the position of the moving limb must be specified on a proprioceptive map using the same system of coordinates as the visuomotor map, so that limb position and target position can coincide. Reference frames can be formed in the physical world, in the sensors themselves, or in the effector system. They do not need to be “frames”; they can be centres of rotation, for instance. Reference frames could also be virtual in the sense of being constructed internally by the brain to perform computations in a topological space, for instance, in which relative positions or motions are the variables that are processed.Less