Topology and Analysis Forum

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Participate in expert discussion on topology and analysis topics. This includes point-set topology, Real, complex, harmonic and functional analysis. Also measure and integration theory.
MHB Limits of Sequences .... Stoll Theorem 2.2.6
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MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (1) ....
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MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.12 ....
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MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (b) ....
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MHB Ordering on the Set of Real Numbers .... Sohrab, Exercise 2.1.10 (c) ....
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I Limits of Sequences .... Bartle & Shebert, Example 3.4.3 (b)
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I Ascending subset sequence with axiom of choice
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MHB Open Sets - Unions and Intersections -
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MHB Limits of Sequences .... Sohrab Exercise 2.2.7
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I Hyperreal Convergence: Is It 0 or Infinitesimal?
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MHB Closed Sets - Sohrab Exercise 2.2.4 - Part 2
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MHB Open and Closed Sets - Sohrab Exercise 2.4.4 - Part 3
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MHB Supremum Property (AoC) .... etc .... Yet a further question/Issue ....
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MHB Supremum Property (AoC) .... etc .... Another question/Issue ....
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MHB Supremum Property (AoC), Archimedean Property, Nested Intervals Theorem ....
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I Supremum Property (AoC) .... etc .... Yet a further question
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I Why Does the Cut of $\mathbb{Q}$ Correspond to $\sqrt{2}$?
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I Supremum Property, Archimedean Property, Nested Intervals
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I Supremum Property (AoC) .... etc .... Another question/Issue
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MHB The Completeness Axiom .... and Sohrab Exercise 2.1.29 ....
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MHB Struggling with Exercise 2.1.40 in Sohrab's Basic Real Analysis?
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MHB Density of Q in R .... Sohrab Theorem 2.1.38 ....
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MHB The AM-GM Inequality - Sohrab Proposition 2.1.25 ....
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I The AM-GM Inequality - Sohrab Proposition 2.1.25 ....
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MHB Proof of Cauchy's Inequality .... Sohrab Proposition 2.1.23 ....
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I Can a tetrahedron have all dihedral angles rational?
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I Differentiating a particular integral (retarded potential)
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I Integers as an Ordered Integral Domain .... Bloch Th. 1.4.6
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B Knot theory: closed loops in n dimensions
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MHB The Integers as an Ordered Integral Domain .... Bloch Theorem 1.4.6 ....
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MHB Some Properties of the Rational Numbers .... Bloch Exercise 1.5.9 (3)
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MHB Sum of a rational number and an irrational number ....
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MHB The Set of Positive Integers as a Copy of the Natural Numbers ....
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I Homotopy Definitions: Homeomorphisms, Homotopies & Retracts
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A Parameterizing the 3-Sphere: 3 Angles?
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I The Set of Positive Integers - a Copy of the Natural Numbers
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I The Well-Ordering Principle for the Natural Numbers ....
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MHB The Well-Ordering Principle for the Natural Numbers
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MHB Properties of "less Than" & "Less Than or Equals" - Bloch Theorem 1.2.9 - Peter
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MHB Addition and Multiplication of Natural Numbers - Bloch Th. 1.2.7 .... ....
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I Properties of 'less than" and "less than or equals"
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I Addition/Multiplication of Natural Numbers - Bloch Th. 1.2.7
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B How Does Group Orbit Theory Relate Torus and Cylinder Structures?
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I Proving non homeomorphism between a closed interval & ##\mathbb{R}##
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A Is This the Correct Way to Attach Möbius Strips to a Torus?
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I Differentiation under the integral in retarded potentials
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I How to show that a 4-polytope is a tiling of the 3-sphere
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B Preferred directions on a torus?
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B Uniqueness of Analytic Functions
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I Solving Complex Integral Paths - Real Line Poles
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I Laplacian of Retarded Potential: How to Derive the Equation Mathematically?
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I Integration of Legendre Polynomials with different arguments
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I Difference between holomoprhic and analytic functions
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I Is it possible to prove that the circle is a manifold using open spheres?
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A Half-face traversal on general polyhedra
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I Merging Two Threads: Complex Integrals & Branch Cuts
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B Real Analysis: Expanding a Function at Different Points
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I Understanding Isolated Singularities in the Residue Theorem
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I Is the Unit Closed Disk Minus the Origin Homeomorphic to the Unit Circle?
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I Jordan Curve Theorem: Exploring Its Complex History
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I Identical zero function in the complex plane
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I Limitations of the Lebesgue Integral
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I Difference between complex and real analysis
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I Pointwise and Uniform Convergence
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I Cauchy's Theorem: Understanding Complex Analysis
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I Proof of Harmonic Function Infinitely Differentiable
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I Boundary and homeomorphism
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MHB Power series and uniform convergence.
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I Learning Complex Integration: Endpoints & Paths
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MHB Proving expression is always positive
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MHB Z²+bz+1=0, where |b|≤2. For which values of b do the following hold?
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B Stereographic projection and uneven scaling
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MHB Why do we have fast convergence?
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I Mapping double cover over a torus
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A Dirac Delta and Residue Calculus
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MHB Sine fourier series with period 1
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I Do these manifolds have a boundary?
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MHB Find cosine series: Period 4 for f(x)
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MHB Can We Prove a Function with Intermediate Value Property is Continuous at x?
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MHB How to prove such value for a derivative?
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I Proving Urysohn's Lemma: Normal vs Perfectly Normal Spaces
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I Is a game situation in chess topologically invariant?
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MHB How Does Sobolev Space Boundedness Relate to Different Norms in $R^n$?
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MHB Need help, are these functions differentiable?
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A Homology groups with points identified
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MHB Series Convergence: Ratio Test & Lim. n→∞
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I Wrong proofs of Riemann hypothesis
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B S u p x i n [ a , b ] | P n ( x ) − f ( x ) | < ϵ
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MHB Proving an Integral with a Direct Proof & Epsilon Argument
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MHB Real Analysis - Prove the Riemann Integral Converges
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MHB The parameter λ in linear integral equations
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MHB Real Analysis - Riemann Integral Proof
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MHB Proving Limits: How to Show x^x Approaches Infinity as x Approaches Infinity?
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MHB How Do You Find the Limit of (e^x - 1) / Sin(x) as x Approaches 0?
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A A pasting lemma counterexample (of sorts)
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I Homotopy Equivalence: Definition & Examples
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A Is this function in the L^2(0,1) space for certain values of C_1 and C_2?
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A What Conditions Determine a Zero Measure Set in Fourier Transforms?
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I Proving Injectivity and Surjectivity: A Fundamental Concept in Function Theory
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I Definition of Topology - What Does {##U_\alpha | \alpha \in I##} Mean?
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Topology and Analysis

Welcome to the Topology forum!

Topology and analysis are two branches of mathematics that investigate the properties of spaces and functions, respectively. Topology is concerned with the study of the properties that remain unchanged under continuous deformations, such as stretching or bending, but not tearing. It explores concepts like continuity, connectedness, and compactness.

Analysis, on the other hand, is focused on understanding the behavior of mathematical functions and sequences. It deals with limits, continuity, differentiation, and integration, providing tools for a detailed examination of functions. Both topology and analysis play integral roles in diverse areas of mathematics and find applications in various scientific and engineering disciplines.

Together, they contribute to a deeper understanding of the structure and properties of mathematical objects.
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