In mathematics, more specifically algebraic topology, a pair is shorthand for an inclusion of topological spaces . Sometimes is assumed to be a cofibration. A morphism from to is given by two maps and such that . A pair of spaces is an ordered pair (X, A) where X is a topological space and A a subspace (with the subspace topology). The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of X by A. Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in . Heuristically, one often thinks of a pair as being akin to the quotient space . There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space to the pair . A related concept is that of a triple (X, A, B), with B ⊂ A ⊂ X. Triples are used in homotopy theory. Often, for a pointed space with basepoint at x0, one writes the triple as (X, A, B, x0), where x0 ∈ B ⊂ A ⊂ X. (Wikipedia).
Definition of a Topological Space
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From playlist Topology
Geogebra Tutorial : Union and Intersection of Sets
Union and intersection of sets can be drawing with geogebra. Please see the video to start how drawing union and intersection of sets. more visit https://onwardono.com
From playlist SET
What is the Symmetric Difference of 2 Sets?
What is the symmetric difference of 2 sets? In this video we go over the symmetric difference of sets, explaining it in a couple ways including what is probably the briefest way. The symmetric difference of two sets A and B is (A union B)-(A intersect B). If you need to know what the defin
From playlist Set Theory
👉 Learn how to define angle relationships. Knowledge of the relationships between angles can help in determining the value of a given angle. The various angle relationships include: vertical angles, adjacent angles, complementary angles, supplementary angles, linear pairs, etc. Vertical a
From playlist Angle Relationships
This video is about topological spaces and some of their basic properties.
From playlist Basics: Topology
Topology (What is a Topology?)
What is a Topology? Here is an introduction to one of the main areas in mathematics - Topology. #topology Some of the links below are affiliate links. As an Amazon Associate I earn from qualifying purchases. If you purchase through these links, it won't cost you any additional cash, b
From playlist Topology
What are parallel lines and a transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
This video is about connectedness and some of its basic properties.
From playlist Basics: Topology
What are the Angle Relationships for Parallel Lines and a Transversal
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal
MAST30026 Lecture 7: Constructing topological spaces (Part 2)
I defined the disjoint union of topological spaces, quotient spaces and the pushout. Lecture notes: http://therisingsea.org/notes/mast30026/lecture7.pdf The class webpage: http://therisingsea.org/post/mast30026/ Have questions? I hold free public online office hours for this class, every
From playlist MAST30026 Metric and Hilbert spaces
Flux periodicity crossover in higher order topological superconductors by Sumathi Rao
DISCUSSION MEETING : EDGE DYNAMICS IN TOPOLOGICAL PHASES ORGANIZERS : Subhro Bhattacharjee, Yuval Gefen, Ganpathy Murthy and Sumathi Rao DATE & TIME : 10 June 2019 to 14 June 2019 VENUE : Madhava Lecture Hall, ICTS Bangalore Topological phases of matter have been at the forefront of r
From playlist Edge dynamics in topological phases 2019
Hermann Schulz-Baldes: Computational K-theory via the spectral localizer.
Talk by Hermann Schulz-Baldes in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on March 24, 2021
From playlist Global Noncommutative Geometry Seminar (Europe)
What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will h
From playlist What is a Manifold?
David Aasen - Topological Defect Networks for Fracton Models - IPAM at UCLA
Recorded 30 August 2021. David Aasen of Microsoft Station Q presents "Topological Defect Networks for Fracton Models" at IPAM's Graduate Summer School: Mathematics of Topological Phases of Matter. Learn more online at: http://www.ipam.ucla.edu/programs/summer-schools/graduate-summer-school
From playlist Graduate Summer School 2021: Mathematics of Topological Phases of Matter
Jay Deep Sau - The return of Majorana : the end of a 75 year-old search
PROGRAM: The ICTS Condensed Matter Programme 2011 Venue: Indian Insitute of Science, Bangalore Date: Friday 09 Dec, 2011 - Thursday 22 Dec, 2011 DESCRIPTION: The ICTS Condensed Matter Programme 2011 (ICMP 2011) consists of a 10 day Winter School (December 9 to December 18) followed by a 4
From playlist The ICTS Condensed Matter Programme 2011
Topological spaces for directed graphs [Henri Riihimäki]
Directed graphs serve as a model for various phenomena in the sciences, for example networks of neurons in the brain or gene regulatory networks. To apply TDA tools we need to construct higher dimensional topological spaces out of directed graphs. In this tutorial we will learn about two s
From playlist Tutorial-a-thon 2021 Fall
Topology 1.4 : Product Topology Introduction
In this video, I define the product topology, and introduce the general cartesian product. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet
From playlist Topology
Laurent Lafforgue - 1/4 Classifying toposes of geometric theories
Course at the school and conference “Toposes online” (24-30 June 2021): https://aroundtoposes.com/toposesonline/ Slides: https://aroundtoposes.com/wp-content/uploads/2021/07/LafforgueSlidesToposesOnline.pdf The purpose of these lectures will be to present the theory of classifying topose
From playlist Toposes online
MathZero, The Classification Problem, and Set-Theoretic Type Theory - David McAllester
Seminar on Theoretical Machine Learning Topic: MathZero, The Classification Problem, and Set-Theoretic Type Theory Speaker: David McAllester Affiliation: Toyota Technological Institute at Chicago Date: May 14, 2020 For more video please visit http://video.ias.edu
From playlist Mathematics
Proving Parallel Lines with Angle Relationships
👉 Learn about converse theorems of parallel lines and a transversal. Two lines are said to be parallel when they have the same slope and are drawn straight to each other such that they cannot meet. In geometry, parallel lines are identified by two arrow heads or two small lines indicated i
From playlist Parallel Lines and a Transversal