Compactness theorems

Kuratowski's intersection theorem

In mathematics, Kuratowski's intersection theorem is a result in general topology that gives a sufficient condition for a nested sequence of sets to have a non-empty intersection. Kuratowski's result is a generalisation of Cantor's intersection theorem. Whereas Cantor's result requires that the sets involved be compact, Kuratowski's result allows them to be non-compact, but insists that their non-compactness "tends to zero" in an appropriate sense. The theorem is named for the Polish mathematician Kazimierz Kuratowski, who proved it in 1930. (Wikipedia).

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Angle of Intersection Between Two Curves

Multivariable Calculus: Find the angle of intersection between the curves r1(t) = (1+t, t, t^3) and r2(t) = (cos(t), sin(t), t^2) at the point (1, 0, 0). For more videos like this one, please visit the Multivariable Calculus playlist at my channel.

From playlist Calculus Pt 7: Multivariable Calculus

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An introduction to the Gromov-Hausdorff distance

Title: An introduction to the Gromov-Hausdorff distance Abstract: We give a brief introduction to the Hausdorff and Gromov-Hausdorff distances between metric spaces. The Hausdorff distance is defined on two subsets of a common metric space. The Gromov-Hausdorff distance is defined on any

From playlist Tutorials

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Yoshihiro Ohnita: Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

An R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form which is a compact embedded totally geodesic Lagrangian submanifold.

From playlist Geometry

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When do vector functions intersect?

Free ebook http://tinyurl.com/EngMathYT Example discussing intersection of curves of two vector functions on one variable.

From playlist Engineering Mathematics

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MATH1081 Discrete Maths: Chapter 5 Question 27 a

This problem is about planar graphs. The theorem mentioned is Fáry's Theorem (1948); see http://bit.ly/1gmUrXT . Presented by Thomas Britz of the School of Mathematics and Statistics, Faculty of Science, UNSW.

From playlist MATH1081 Discrete Mathematics

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The Homework Problem That Started as a Phd Thesis: 14 set theorem

In a handful of introductory topology textbooks, Kuratowski's 14 set theorem is given as an exercise despite it being one of the results proven as a part of his phd thesis in 1922. This homework problem that started out as a phd thesis is not an easy exercise if you don't know how to think

From playlist The New CHALKboard

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A Classification of Planar Graphs - A Proof of Kuratowski's Theorem

A visually explained proof of Kuratowski's theorem, an interesting, important and useful result classifying "planar" graphs. Proof adapted from: http://math.uchicago.edu/~may/REU2017/REUPapers/Xu,Yifan.pdf and: https://www.math.cmu.edu/~mradclif/teaching/228F16/Kuratowski.pdf Also check

From playlist Summer of Math Exposition Youtube Videos

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Ex: Determine the Point of Intersection of a Plane and a Line.

This video explains how to determine the intersection point of a plane and a line. http://mathispower4u.com

From playlist Equations of Planes and Lines in Space

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Planar Graphs - Numberphile

Featuring Professor Maria Chudnovsky from Princeton University - see part two about her work on Perfect Graphs - https://youtu.be/C4Zr4cOVm9g More links & stuff in full description below ↓↓↓ Correction at 13:58 - remove the word "not". Professor Chudnovsky's webpage: https://web.math.pri

From playlist Women in Mathematics - Numberphile

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Find the Angle of Intersection of Two Space Curves Given As Vector Functions

This video illustrates and explains how to determine the acute angle of intersection between two space curves given as vector valued functions. http://mathispower4u.com

From playlist Vector Valued Functions

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Graph Theory: 61. Characterization of Planar Graphs

We have seen in a previous video that K5 and K3,3 are non-planar. In this video we define an elementary subdivision of a graph, as well as a subdivision of a graph. We then discuss the fact that if a graph G contains a subgraph which is a subdivision of a non-planar graph, then G is non-

From playlist Graph Theory part-10

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Set Theory (Part 3): Ordered Pairs and Cartesian Products

Please feel free to leave comments/questions on the video and practice problems below! In this video, I cover the Kuratowski definition of ordered pairs in terms of sets. This will allow us to speak of relations and functions in terms of sets as the basic mathematical objects and will ser

From playlist Set Theory by Mathoma

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Math 131 092116 Properties of Compact Sets

Properties of compact sets. Compact implies closed; closed subsets of compact sets are compact; collections of compact sets that satisfy the finite intersection property have a nonempty intersection; infinite subsets of compact sets must have a limit point; the infinite intersection of ne

From playlist Course 7: (Rudin's) Principles of Mathematical Analysis

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Ulrich Bauer (3/4/22): Gromov hyperbolicity, geodesic defect, and apparent pairs in Rips filtrations

Motivated by computational aspects of persistent homology for Vietoris-Rips filtrations, we generalize a result of Eliyahu Rips on the contractibility of Vietoris-Rips complexes of geodesic spaces for a suitable parameter depending on the hyperbolicity of the space. We consider the notion

From playlist Vietoris-Rips Seminar

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Rade Zivaljevic (6/27/17) Bedlewo: Topological methods in discrete geometry; new developments

Some new applications of the configurations space/test map scheme can be found in Chapter 21 of the latest (third) edition of the Handbook of Discrete and Computational Geometry [2]. In this lecture we focus on some of the new developments which, due to the limitations of space, may have b

From playlist Applied Topology in Będlewo 2017

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Graph Theory: 62. Graph Minors and Wagner's Theorem

In this video, we begin with a visualisation of an edge contraction and discuss the fact that an edge contraction may be thought of as resulting in a multigraph or simple graph, depending on the application. We then state the definition a contraction of edge e in a graph G resulting in a

From playlist Graph Theory part-10

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L. Meersseman - Kuranishi and Teichmüller

Abstract - Let X be a compact complex manifold. The Kuranishi space of X is an analytic space which encodes every small deformation of X. The Teichmüller space is a topological space formed by the classes of compact complex manifolds diffeomorphic to X up to biholomorphisms smoothly isotop

From playlist Ecole d'été 2019 - Foliations and algebraic geometry

Related pages

Compact space | Complete metric space | General topology | Mathematics | Measure of non-compactness | Cantor's intersection theorem | Intersection | Closed set