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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

## 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

## 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

## 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 *S*^{2} 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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