Topological spaces

Partition topology

In mathematics, the partition topology is a topology that can be induced on any set X by partitioning X into disjoint subsets P; these subsets form the basis for the topology. There are two important examples which have their own names: * The odd–even topology is the topology where and Equivalently, . * The deleted integer topology is defined by letting and . The trivial partitions yield the discrete topology (each point of X is a set in P, so ) or indiscrete topology (the entire set X is in P, so ). Any set X with a partition topology generated by a partition P can be viewed as a pseudometric space with a pseudometric given by: This is not a metric unless P yields the discrete topology. The partition topology provides an important example of the independence of various separation axioms. Unless P is trivial, at least one set in P contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, X is regular, completely regular, normal and completely normal. X/P is the discrete topology. (Wikipedia).

Video thumbnail

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

Video thumbnail

Topology 1.6 : Metric Topology

In this video, I introduce the metric topology, and introduce how the topologies it generates align with the standard topologies on Euclidean space. Email : fematikaqna@gmail.com Subreddit : https://www.reddit.com/r/fematika Code : https://github.com/Fematika/Animations Notes : None yet

From playlist Topology

Video thumbnail

Partitions of a Set | Set Theory

What is a partition of a set? Partitions are very useful in many different areas of mathematics, so it's an important concept to understand. We'll define partitions of sets and give examples in today's lesson! A partition of a set is basically a way of splitting a set completely into disj

From playlist Set Theory

Video thumbnail

Definition of a Topological Space

Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Definition of a Topological Space

From playlist Topology

Video thumbnail

Topology 1.3 : Basis for a Topology

In this video, I define what a basis for a topology is. Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animations Notes : None yet

From playlist Topology

Video thumbnail

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

Video thumbnail

Topology 1.7 : More Examples of Topologies

In this video, I introduce important examples of topologies I didn't get the chance to get to. This includes The discrete and trivial topologies, subspace topology, the lower-bound and K topologies on the reals, the dictionary order, and the line with two origins. I also introduce (again)

From playlist Topology

Video thumbnail

Topology: Metric Spaces

This video is about metric spaces and some of their basic properties.

From playlist Basics: Topology

Video thumbnail

What is a closed set ?

I define closed sets, an important notion in topology and analysis. It is defined in terms of limit points, and has a priori nothing to do with open sets. Yet I show the important result that a set is closed if and only if its complement is open. More topology videos can be found on my pla

From playlist Topology

Video thumbnail

Bourbaki - 16/01/2016 - 1/4 - Damien GABORIAU

Damien GABORIAU — Entropie sofique [d'après L. Bowen, D. Kerr et H. Li] L’entropie fut introduite en systèmes dynamiques par A. Kolmogorov. Initialement focalisée sur les itérations d’une transformation préservant une mesure finie, la notion fut peu à peu généralisée, jusqu’à embrasser l

From playlist Bourbaki - 16 janvier 2016

Video thumbnail

The role of topology and compactness (...) - CEB T2 2017 - Varadhan - 3/3

S.R.S. Varadhan (Courant Institute) - 09/06/2017 The role of topology and compactness in the theory of large deviations When a large deviation result is proved there is some topology involved in the statement because it affects the class of sets for which the estimates hold. Often the cho

From playlist 2017 - T2 - Stochastic Dynamics out of Equilibrium - CEB Trimester

Video thumbnail

Higher Topological Charge Effects in QCD and Beyond by Fabian Rennecke

DISCUSSION MEETING TOPOLOGICAL ASPECTS OF STRONG CORRELATIONS AND GAUGE THEORIES (ONLINE) ORGANIZERS: Rob Pisarski (Brookhaven National Laboratory, USA), Sumathi Rao (HRI, India), Soeren Schlichting (Bielefeld University, Germany) and Sayantan Sharma (IMSc, India) DATE: 06 September 202

From playlist Topological aspects of strong correlations and gauge theories (ONLINE)

Video thumbnail

Symmetries in QFT and their Relationship with Category Theory (Lecture 2) by Lakshya Bhardwaj

INFOSYS-ICTS STRING THEORY LECTURES SYMMETRIES IN QFT AND THEIR RELATIONSHIP WITH CATEGORY THEORY SPEAKER: Lakshya Bhardwaj (Mathematical Institute, University of Oxford) DATE : 10 October 2022 to 12 October 2022 VENUE: Madhava Lecture Hall (Hybrid) Lecture 1: 10 October 2022 at 3:30 pm

From playlist Infosys-ICTS String Theory Lectures

Video thumbnail

N=2* SU(2) Supersymmetric Yang-Mills Theory and Four-Manifold Invariants - Gregory Moore

High Energy Theory Seminar N=2* SU(2) Supersymmetric Yang-Mills Theory and Four-Manifold Invariants Speaker: Gregory Moore Affiliation: Rutgers University Date: March 15, 2021 For more video please visit http://video.ias.edu

From playlist IAS High Energy Theory Seminar

Video thumbnail

Lecture 3: VOA[M4] (Lecture 2) by Sergei Gukov

Program: Quantum Fields, Geometry and Representation Theory ORGANIZERS : Aswin Balasubramanian, Saurav Bhaumik, Indranil Biswas, Abhijit Gadde, Rajesh Gopakumar and Mahan Mj DATE & TIME : 16 July 2018 to 27 July 2018 VENUE : Madhava Lecture Hall, ICTS, Bangalore The power of symmetries

From playlist Quantum Fields, Geometry and Representation Theory

Video thumbnail

Equidistribution of Measures with High Entropy for General Surface Diffeomorphisms by Omri Sarig

PROGRAM : ERGODIC THEORY AND DYNAMICAL SYSTEMS (HYBRID) ORGANIZERS : C. S. Aravinda (TIFR-CAM, Bengaluru), Anish Ghosh (TIFR, Mumbai) and Riddhi Shah (JNU, New Delhi) DATE : 05 December 2022 to 16 December 2022 VENUE : Ramanujan Lecture Hall and Online The programme will have an emphasis

From playlist Ergodic Theory and Dynamical Systems 2022

Video thumbnail

Off-shell Partition Functions in 3d Gravity - Lorenz Eberhardt

IAS Physics Group Meeting Topic: Off-shell Partition Functions in 3d Gravity Speaker: Lorenz Eberhardt Affiliation: Member, School of Natural Sciences, IAS Date: May 25, 2022 I will discuss partition functions in three-dimensional quantum gravity with negative cosmological constant in ca

From playlist Physics Group Meeting

Video thumbnail

Karen VOGTMANN - Spaces of Graphs, Tori and Other Flat Gamma-complexes

Spaces of finite graphs play a key role in perturbative quantum field theory, but also in many other areas of science and mathematics. Among these is geometric group theory, where they are used to model groups of automorphism of free groups. Graphs can be thought of as 1-dimensional flat m

From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

Video thumbnail

Steffen Borgwardt: The role of partition polytopes in data analysis

The field of optimization, and polyhedral theory in particular, provides a powerful point of view on common tasks in data analysis. In this talk, we highlight the role of the so-called partition polytopes and their studies in clustering and classification. The geometric properties of parti

From playlist Workshop: Tropical geometry and the geometry of linear programming

Related pages

Pseudometric space | Topological space | Urysohn and completely Hausdorff spaces | Regular space | Counterexamples in Topology | Kolmogorov space | Mathematics | Normal space | Partition of a set | T1 space | Hausdorff space