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Least-upper-bound property
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1	1. Thank you for the invitation. 2. I gave an invited talk at CCA in Hagen, I think it was in 2008, where I spoke about real numbers. Today I am speaking about real numbers again. You know what to expect the next time. Add to Reading List Source URL: math.andrej.com Language: English - Date: 2016-06-16 04:03:45 Elementary mathematics Metric geometry Order theory Real numbers Real algebraic geometry Complete metric space Cauchy sequence Construction of the real numbers Augustin-Louis Cauchy Sequence Least-upper-bound property Completeness
2	In defence of Dedekind and Heine–Borel Paul Taylor Third Workshop on Formal Topology Padova, mercoled`ı, il 9 Maggio 2007 Add to Reading List Source URL: www.paultaylor.eu Language: English - Date: 2009-02-12 12:35:14 Metric geometry Real analysis Least-upper-bound property Order theory Cauchy sequence Real number Mathematical logic Complete metric space Compact space Mathematics Mathematical analysis Topology
3	The Dedekind Reals in Abstract Stone Duality Andrej Bauer and Paul Taylor 3 June 2009 Abstract Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other con Add to Reading List Source URL: www.paultaylor.eu Language: English - Date: 2009-06-03 17:25:38 General topology Order theory Real analysis Calculus Measure theory Least-upper-bound property Compact space Intermediate value theorem Continuous function Mathematical analysis Topology Mathematics
4	Interval Analysis Without Intervals Paul Taylor 20 February 2006 Abstract We argue that Dedekind completeness and the Heine–Borel property should be seen as part of the “algebraic” structure of the real line, along Add to Reading List Source URL: www.paultaylor.eu Language: English - Date: 2009-02-12 13:25:28 Order theory Real analysis Differential calculus Elementary mathematics Continuous function Dedekind cut Fundamental theorem of calculus Integral Least-upper-bound property Mathematical analysis Mathematics Functions and mappings
5	2001 Paper 1 Question 8 Discrete Mathematics Let (A, 6A ) and (B, 6B ) be partially ordered sets. (a) Define the product order on A×B and prove that it is a partial order. [4 marks] Add to Reading List Source URL: www.cl.cam.ac.uk Language: English - Date: 2014-06-09 10:17:39 Abstraction Lattice Supremum Partially ordered set Upper and lower bounds Infimum Ordinal optimization Least-upper-bound property Order theory Mathematics Structure
6	INDUCTION AND COMPLETENESS IN ORDERED SETS PETE L. CLARK Abstract. We deﬁne an inductive subset of an ordered set and show that an ordered set X is Dedekind complete iﬀ the only inductive subset of X is X itself. Thi Add to Reading List Source URL: math.uga.edu Language: English - Date: 2013-09-10 16:37:35 Abstraction Linear continuum Least-upper-bound property Supremum Order topology Filter Construction of the real numbers Mathematics Order theory Structure
7	Lecture 3 Limits In the previous lecture, we used the least upper bound property of the real numbers to define the basic arithmetic operations of addition and multiplication. In effect, this involved finding sequences wh Add to Reading List Source URL: math.caltech.edu Language: English - Date: 2013-10-14 09:59:36 Real algebraic geometry Real number Multiplication Triangle inequality Least-upper-bound property Limit superior and limit inferior Series Mathematics Mathematical analysis Real analysis
8	Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem The purpose of this lecture is more modest than the previous ones. It is to state certain conditions under which we are guaranteed that limits of Add to Reading List Source URL: math.caltech.edu Language: English - Date: 2013-08-13 12:27:41 Real analysis Metric geometry Real numbers Cauchy sequence Topology Least-upper-bound property Bolzano–Weierstrass theorem Series Augustin-Louis Cauchy Mathematical analysis Mathematics Elementary mathematics
9	2 Sequences and Series In this chapter we will study two related questions. Given an inﬁnite collection X of numbers, which can be taken to be rational, real or complex, the ﬁrst question is to know if there is a li Add to Reading List Source URL: www.math.caltech.edu Language: English - Date: 2010-10-16 18:41:12 Metric geometry Elementary mathematics Real analysis Real numbers Convergence Cauchy sequence Series Least-upper-bound property Complete metric space Mathematical analysis Mathematics Geometry
10	Lecture 2: The real numbers The purpose of this lecture is for us to develop the real number system. This might seem like a very strange thing for us to be doing. It must seem to you that you have been studying real numb Add to Reading List Source URL: math.caltech.edu Language: English - Date: 2013-08-06 12:25:56 Elementary mathematics Field theory Real numbers Real algebraic geometry Least-upper-bound property Multiplication Supremum Number system Well-order Mathematics Abstract algebra Order theory

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