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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

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0011
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

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


Topology

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0006
Subject:
Economics and Finance, Econometrics

This chapter discusses topological spaces and associated concepts, including first‐ and second‐countability, compactness, and separation properties. Weak topologies are defined. The case of product ... More


SPACES AS COMONOIDS

A. Bucalo and G. Rosolini

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

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0012
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

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


Essentials of General Topology

Adel N. Boules

in Fundamentals of Mathematical Analysis

Published in print:
2021
Published Online:
May 2021
ISBN:
9780198868781
eISBN:
9780191905254
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198868781.003.0005
Subject:
Mathematics, Analysis, Applied Mathematics

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


Overview

Loring W. Tu

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0001
Subject:
Mathematics, Educational Mathematics

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


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

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0010
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

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


Adic spaces II

Peter Scholze and Jared Weinstein

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691202082
eISBN:
9780691202150
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691202082.003.0003
Subject:
Mathematics, Geometry / Topology

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


Equivariant Cohomology of S2 Under Rotation

Loring W. Tu

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0007
Subject:
Mathematics, Educational Mathematics

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


Topology

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
2021
Published Online:
November 2021
ISBN:
9780192844507
eISBN:
9780191927201
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192844507.003.0006
Subject:
Economics and Finance, Econometrics

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


Symmetric Markovian Semigroups and Dirichlet Forms

Zhen-Qing Chen and Masatoshi Fukushima

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

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691136059
eISBN:
9781400840564
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691136059.003.0001
Subject:
Mathematics, Probability / Statistics

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


Diamonds associated with adic spaces

Peter Scholze and Jared Weinstein

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691202082
eISBN:
9780691202150
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691202082.003.0010
Subject:
Mathematics, Geometry / Topology

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


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

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159294
eISBN:
9781400851317
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159294.003.0015
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

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


Transcendental Deduction II: The B Edition Transcendental Deduction

Thomas C. Vinci

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

Published in print:
2014
Published Online:
October 2014
ISBN:
9780199381166
eISBN:
9780199381234
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199381166.003.0008
Subject:
Philosophy, History of Philosophy, Metaphysics/Epistemology

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


Homotopy Groups and CW Complexes

Loring W. Tu

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0002
Subject:
Mathematics, Educational Mathematics

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


The space of stably dominated types

Ehud Hrushovski and François Loeser

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

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0003
Subject:
Mathematics, Geometry / Topology

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


v-sheaves associated with perfect and formal schemes

Peter Scholze and Jared Weinstein

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

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691202082
eISBN:
9780691202150
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691202082.003.0018
Subject:
Mathematics, Geometry / Topology

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


An equivalence of categories

Ehud Hrushovski and François Loeser

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

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691161686
eISBN:
9781400881222
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161686.003.0013
Subject:
Mathematics, Geometry / Topology

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


Frames of reference used in goal-directed arm movement

Sylvie Vanden Abeele, Marc Crommelinck, and André Roucoux

in Multisensory Control of Movement

Published in print:
1993
Published Online:
March 2012
ISBN:
9780198547853
eISBN:
9780191724268
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198547853.003.0181
Subject:
Neuroscience, Sensory and Motor Systems

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


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