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• MAT5203
  Introduction to Lebesgue measure and integration. Sample textbook: Real Analysis by H. L. Royden. || [[Review of the field and completeness axions]]
  549 bytes (62 words) - 14:06, 18 March 2023
• MAT3333
  MAT 3333 Fundamentals of Analysis and Topology. MAT 3333 Fundamentals of Analysis and Topology.
  4 KB (586 words) - 15:57, 25 March 2023
• Absolute Value and the Real Line
  == The Absolute Value of a Real Number == ...f <math>a</math> is a real number, then we define the '''absolute value of the number <math>a</math>''' denoted <math>\mid a \mid</math> or <math>\mathrm{
  11 KB (1,976 words) - 13:18, 27 November 2021
• Subspaces of Metric Spaces
  = Subspace topology= ...he '''relative topology''', or the '''induced topology''', or the '''trace topology''').
  11 KB (1,955 words) - 14:30, 23 January 2022
• Topological Spaces
  ...oints, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. ...e most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Other spaces, such as Euclidean spac
  19 KB (3,169 words) - 09:58, 29 October 2021
• Open Subsets
  ...red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.]] ...a metric space—that is, when a distance function is defined—open sets are the sets that, with every point {{mvar|P}}, contain all points that are suffici
  24 KB (4,147 words) - 16:13, 28 October 2021
• MAT4223
  [[Uniform Convergence of Sequences of Functions]] * [[The Cauchy Criterion for Convergence]] <!-- 3213-3.5 -->
  9 KB (1,029 words) - 16:37, 29 January 2022
• MAT3223
  ...213. An introduction to complex variables, including elementary functions, line integrals, power series, residues and poles, and conformal mappings. Genera * Complex numbers and the complex plane.
  6 KB (816 words) - 11:00, 24 March 2023
• Totally Bounded Metric Spaces
  ...very fixed "size" (where the meaning of "size" depends on the structure of the ambient space.) ...so used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
  9 KB (1,582 words) - 20:18, 13 November 2021
• Separation Properties
  ...notion of connected spaces (and their connected components) as well as to the separation axioms for topological spaces. There are various ways in which two subsets of a topological space ''X'' can be considered to be separated.
  6 KB (1,047 words) - 21:31, 29 January 2022
• Distance Functions, Metrics
  ...ic, the green path has length <math>6 \sqrt{2} \approx 8.49</math>, and is the unique shortest path.]] ...not all topologies can be generated by a metric. A topological space whose topology can be described by a metric is called metrizable.
  16 KB (2,652 words) - 14:31, 29 October 2021
• MAT4273
  The textbook(s) for this course is [1] Undergraduate Topology, by R. H. Kasriel; topics from [2] Introduction
  12 KB (1,471 words) - 15:09, 23 January 2022
• Critical Points of a Function
  ...|The abscissae ("x-coordinates") of the red circles are stationary points; the blue squares are inflection points.]] '''Critical point''' is a wide term used in many branches of mathematics.
  16 KB (2,821 words) - 17:17, 14 November 2021
• Algebraic Structure of the Real Numbers
  ...ing the symbol '''R''' or <math>\mathbb{R}</math> and is sometimes called "the reals". ...of the complex plane, and the real numbers can be thought of as a part of the complex numbers.
  28 KB (4,634 words) - 16:18, 25 October 2021
• MAT3213
  ...gence; functions and function limits. Introduction to topology of the real line. Emphasis on theorem proving. ''Sample textbook'': Introduction to Real Analysis by Bartle and Sherbert
  8 KB (729 words) - 20:56, 24 March 2023
• Continuous Mappings Between Metric Spaces
  ==Types of maps between metric spaces== The map <math>f\,\colon M_1\to M_2</math> is continuous
  4 KB (724 words) - 13:10, 23 January 2022
• Metric Spaces
  ...ce'' between any two members of the set, which are usually called points. The metric satisfies a few simple properties. Informally: ...math>B</math> is zero if and only if <math>A</math> and <math>B</math> are the same point,
  29 KB (5,181 words) - 22:13, 26 October 2021
• The Hilbert Space L2 and the Hilbert Cube
  ==Definition and illustration of Hilbert Space== ...x'''}} and {{math|'''y'''}} are represented in Cartesian coordinates, then the dot product is defined by
  17 KB (2,752 words) - 14:34, 10 November 2021
• Metric spaces
  <noinclude>{{navigation|Book=Topology ...esque and accessible; it will subsequently lead us to the full abstraction of a topological space.
  37 KB (6,977 words) - 16:54, 28 September 2021
• Continuous Functions
  ...tions as well, to provide some remarks of common functions that do not fit the bill. ...s on the domain <math>\R\setminus \{0\}</math>, but is not continuous over the domain <math>\R</math> because it is undefined at <math>x=0</math>]]
  21 KB (3,737 words) - 23:56, 5 November 2021
• Real Numbers:Irrational
  [[File:Square root of 2 triangle.svg|right|thumb|240px|The number <math> \sqrt{2} </math> is irrational.]] ...express the lengths of both of the two given segments as integer multiples of itself.
  14 KB (2,317 words) - 15:55, 21 October 2021
• A Topology Given By A Metric
  ...nfty)</math>, is a function which is called the ''metric'' which satisfies the requirement that for all <math>a,b,c\in X</math>: ...tate <math>(X,d)</math> when we talk of a metric space <math>X</math> with the metric <math>d</math>.
  36 KB (6,841 words) - 16:57, 13 November 2021
• Real Numbers (Rational vs. Irrational Numbers)
  ...es including the integers (<math>\mathbb{Z}</math>), which in turn include the natural numbers (<math>\mathbb{N}</math>)]] ...to as "'''the rationals'''", the '''field of rationals''' or the '''field of rational numbers''' is usually denoted by a boldface {{math|'''Q'''}} ; it
  29 KB (4,667 words) - 17:58, 11 December 2021
• Real Numbers:Intervals
  [[File:Interval0.png|thumb|400px|The addition ''x'' + ''a'' on the number line. All numbers greater than ''x'' and less than ''x'' + ''a'' fall within tha ...s, the set of positive real numbers, the empty set, and any singleton (set of one element).
  17 KB (2,805 words) - 14:58, 6 November 2021
• The Definition of the Limit of a Function
  ...is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. ...o ''p'' are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''.
  28 KB (4,901 words) - 16:24, 12 January 2022
• Rings
  ...ns satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, ...iplicative identity element. (Some authors use the term "ring" to refer to the more general structure that omits this last requirement.)
  71 KB (12,100 words) - 16:43, 19 December 2021