What is Topological: Definition and 264 Discussions

In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

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  1. S

    I Topological phase transitions for the whole Universe...?

    Physicist Grigory Volovik has put forward some ideas about the universe undergoing a topological phase transition (especially in the early stages of the universe). He published a book called "*The Universe in a Helium Droplet*" where he explained his ideas. You can find a brief discussion here...
  2. S

    I Inhomogeneities and topological defects in cosmology...

    I have heard that some types of inhomogeneties and topological defects (like cosmic strings) in cosmology have been proposed to be able to break fundamental symmetries of nature such as the Poincaré, Lorentz, diffeomorphism CPT, spatial/time translational...etc symmetries... However, I have not...
  3. A

    Solid State Best resources to learn topological condensed matter in 2022?

    Hello, I was not sure whether this should belong to this section or the condensed matter section. I am wondering if after about 15 years in research in topological condensed matter, there exist well-recognized references for beginners in the topic. Books or courses but also review articles...
  4. D

    Solid State Texts on Topological Effects/Phases in Materials

    I am looking to learn about these topological effects or phases in solids. More specifically, I'm trying to find a set of lecture notes or a textbook or some other text that do not shy away from discussing homotopy classes and the application algebraic topology to describe these materials. I...
  5. hagopbul

    I Topological question from Ashcroft-Mermin

    Hello : doing some reading in physics and some of it is in solid state physics , in Ashcroft- mermin book chapter 2 page 33 you read " Thus if our metal is one dimensional we would simply replace the line from 0 to L to which the electron were confined by a circle of circumference L. In...
  6. W

    A Topological Definition of Holes

    Is it reasonable to define the n-dimensional holes of a topological space X as the non-zero Homology/Homotopy classes of X? Can we read these as obstructions to continuously shrinking a simple closed curve * to a point within the space? *I understand this is what we mean by a cycle.
  7. K

    A Introduction to topological field theory?

    Hi! I have been looking at differential forms, manifolds and de Rham cohomology. Now I'm trying to figure out the connection from cohomology and equations of motions and topological field theory. Problem is that I am only looking at abelian field theories and I only find introductions into...
  8. L

    A What Topological Vector Spaces have an uncountable Schauder basis?

    Let ##P## be an uncountable locally finite poset, let ##F## be a field, and let ##Int(P)=\{[a,b]:a,b\in P, a\leq b\}##. Then the incidence algebra $I(P)$ is the set of all functions ##f:P\rightarrow F##, and it's a topological vector space over ##F## (a topological algebra in fact) with an...
  9. T

    B Applications of topological spaces not homeomorphic to R^n in physics

    Hello. So, the question is do you know any applications of topological spaces which are not homeomorphic to R^n in physics? Motivation for the question i am making: as i think if a topological space is homeomorphic to R^n then differential calculus is allowed on it. Modern physics uses i think...
  10. W

    B Topological Gravity Explained: Cumrum Vafa's Model

    I have heard Cumrum Vafa one of the leading figure sin string theory proposed a model of the early universe using something called "topological gravity". Can anyone explain what this refers to? A paper can be found here: https://arxiv.org/pdf/2009.10077.pdf
  11. F

    A Are topological defects such as domain walls really discontinuous?

    Are topological defects such as domain walls really discontinuous (implying infinite slopes of the fields) or only approximately ?
  12. L

    A Topological Basis in l2 Space: Why?

    Why in ##l_2## space basis ##|1 \rangle=[1 \; 0 \; 0 ...]^{\mathsf{T}}##, ##|2 \rangle=[0 \; 1 \; 0 ...]^{\mathsf{T}}##, ##|3 \rangle=[0 \; 0 \; 1 \; 0...]^{\mathsf{T}}##... is called topological basis?
  13. T

    I Topological Defects: Can They Induce Phase Transitions?

    I'm curious to know about topological defects and whether they can induce phase transitions in condensed matter physics. Can anyone entertain me?
  14. JackHolmes

    A Help with the Derrick scaling argument and topological solitons

    I have been reading Manton & Sutcliffe for some time now and can't quite wrap my head around something. If you take the Hopf invariant N of a topological soliton ϕ then its Skyrme-Faddeev energy (which I hope I've gotten right up to some constants) E=∫∂iϕ⋅∂iϕ+(∂iϕ×∂jϕ)⋅(∂iϕ×∂jϕ) d3x satisfies...
  15. Replusz

    30-day Topological Quantum challenge

    Hey everyone, This is more of a motivational thread, and of course if anyone wants to join in, please do! Any comments are welcome. It's also fine if no one comments. Maybe don't remove the thread though please. I hope this might be useful later on for others as motivation. So the challenge is...
  16. B

    Topological insulators and their optical properties

    I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an...
  17. Math Amateur

    I Interior and Closure in a Topological Space .... .... remark by Willard

    I am reading Stephen Willard: General Topology ... ... and am currently focused on Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to fully understand a result or formula given by Willard concerning a link between...
  18. Math Amateur

    I Closure in a Topological Space .... Willard, Theorem 3.7 .... ....

    I am reading Stephen Willard: General Topology ... ... and am currently reading Chapter 2: Topological Spaces and am currently focused on Section 1: Fundamental Concepts ... ... I need help in order to fully understand an aspect of the proof of Theorem 3.7 ... ..Theorem 3.7 and its proof...
  19. Math Amateur

    I Convergence in Topological Spaces .... Singh, Example 4.1.1 .... ....

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ... I need help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows: In the above example from Singh we read the following: "...
  20. Math Amateur

    I Limit Points & Closure in a Topological Space .... Singh, Theorem 1.3.7

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 1, Section 1.2: Topological Spaces ... I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to...
  21. Math Amateur

    I Definitions of Continuity in Topological Spaces ....

    I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...
  22. Math Amateur

    MHB Compact Topological Spaces .... Stromberg, Theorem 3.36 .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows: In the...
  23. Math Amateur

    MHB Compact Topological Spaces .... Stromberg, Example 3.34 (c) .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand an aspect of Example 3.34 (c) on page 102 ... ... Examples 3.34 (plus some relevant definitions ...)...
  24. cianfa72

    I Full-cone as topological space

    Hello, consider a full-cone (let me say a cone including bottom half, upper half and the vertex) embedded in ##E^3##. We can endow it with the topology induced by ##E^3## defining its open sets as the intersections between ##E^3## open sets (euclidean topology) and the full-cone thought itself...
  25. S

    I On the topological proof of the Fundamental Theorem of Algebra

    Sorry for the misspelling, but this forum doesn't allow enough characters for the title. The title should be: For the topological proof of the Fundamental Theorem of Algebra, what is the deal when the roots are at the same magnitude, either at different complex angles, or repeated roots? I...
  26. J

    A Topological Pseudodefects of a Supersymmetric SO(10) Model

    Here they discuss supersymmetry, SO(10) GUT, inflation, and domain walls. Topological Pseudodefects of a Supersymmetric SO(10) Model and Cosmology https://journals.aps.org/prd/pdf/10.1103/PhysRevD.98.063523 Topological pseudodefects of a supersymmetric SO(10)model and cosmology Ila Garg and...
  27. H

    A Is there a topological insulator without Spin Orbit Coupling (SOC)?

    There are some famous materials is determined as TI induced by SOC, like graphene and so on. But from some formula, for instance, Kane-Fu formula, they just need parities to get Z2 number. So I wonder if there is a known TI with weak soc.
  28. E

    A Surface states of 3D topological insulators

    I have a question (more like a curiosity) related to three-dimensional topological insulators, which support Dirac-like states at their surfaces. From the theory, it is well known that these states are immune to scattering from non-magnetic impurities, i.e. impurities that do not break...
  29. S

    A Topological Phases: Understanding Kitaev's Paper

    Dear All I am trying to understand the following paper for Kitaev :" Periodic table for topological insulators and superconductors", But i am founding it so hard. Can anyone help me to understand it? Thank you.
  30. Wendel

    B Nesting of 2-Spheres & 2-Tori in Topological Spaces

    Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?
  31. nightingale123

    Topology: Understanding open sets

    Homework Statement We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this ##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## ##a)## Show that ##\tau## satisfies that axioms for...
  32. nightingale123

    Finding homeomorphism between topological spaces

    Homework Statement show that the two topological spaces are homeomorphic. Homework Equations Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them The Attempt at a Solution I have tried proving that these two spaces are homeomorphic...
  33. J

    Solid State Sources to learn about Topological Superconductivity

    Greetings. Does anyone know about any good places to learn about topological superconductivity from? Thanks in advance!
  34. J

    Solid State Books on Topological Insulators

    Hello. Do you know of any good material on topological insulators like books, review papers etc? I would prefer something more oriented towards theoretical physics(because I know that there are reviews out there that are purely experimental). Thank you!
  35. J

    Solid State Books: Weyl semimetals, Topological Insulators

    Hello! What are some good sources(preferably textbooks) to learn about Weyl semimetals? I also want some sources to learn about topological insulators and anything containing the Integer Quantum Hall effect would be great. As an aside, if you have any good book on theoretical condensed matter...
  36. N

    A Topological Quantum Field Theory: Help reading a paper

    https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space ##Y## and a spacetime ##X## the author talks about the associated Quantum Hilbert Spaces...
  37. T

    A Self-Teaching Topological Insulators/Phases/Symmetry

    Hey I am a recent graduate with a B.S. in physics and mathematics. The highest physics class I officially took is Quantum Mechanics. I am very interested in learning about topological insulators, topological phases, and topological symmetry, but when I look at papers in the field on Arxiv I can...
  38. FallenApple

    Studying How to learn Topological Data Analysis

    It's a really interesting idea. I think I want to eventually add this to my toolbox. But how? I've heard it uses ideas from algebraic topology. But how much of theoretical topology do I actually need to learn? Are proofs important? I don't care about developing the algorithms from scratch. I...
  39. Julio1

    MHB Convergence in topological space

    Let $(X,\tau)$ an topological space. Show that $x_n\to_{n\to \infty} x$ if and only if $d(x_n,x)\to_{n\to \infty} 0.$ Hello, any idea for begin? Thanks.
  40. N

    Topological Basis Homework: If-Then Conditions

    Homework Statement Let (χ,τ) be a topological space and β be a collection of subsets of χ. Then β is a basis for τ if and only if: 1. β ⊂ τ 2. for each set U in τ and point p in U there is a set V in β such that p ∈ V ⊂ U. 2. Relevant definitions Let τ be a topology on a set χ and let β ⊂ τ...
  41. A

    A Are all edge states topological?

    Hey am new to this forum but I have a question regarding topologically protected states.. Let's suppose we have a 1D gapped system divided two to distinct regions that have different periodicity or different properties and that at the centre, where the two regions 'meet' states appear in the...
  42. N

    A Gauge Theory: Principal G Bundles

    I've been studying TQFT and gauge theory. Dijkgraaf-Witten theory in particular. One learns that a topological field theory applied to a manifold outputs the number of principal G bundles of a manifold. My question is for the Physicists in the room, why do you want to know the number of...
  43. S

    Solid State Topological Superconductors -- Looking for introductory textbooks or study materials

    Dear All I am trying to study Topological superconductors but i have no idea about it. Can anyone suggest me an introductory book to start with.
  44. T

    Booster transformer topology for transmission application

    Hi, Regarding transmission line booster transformer topologies I'm curious as to what would happen over all the possible permutations. I refer you to the following: http://top10electrical.blogspot.com.au/2015/03/booster-transformer.html I presume that in real life the booster transformers...
  45. A

    A ##2+1##-D Einstein gravity is topological and only non-trivial locally

    ##2+1##-dimensional Einstein gravity has no local degrees of freedom. This can be proved in two different ways: 1. In ##D##-dimensional spacetime, a symmetric metric tensor appears to have ##\frac{D(D+1)}{2}## degrees of freedom satisfying ##\frac{D(D+1)}{2}## apparently independent Einstein...
  46. Wendel

    Path-connectedness for finite topological spaces

    Homework Statement I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? Homework Equations how can I...
  47. S

    Is 1 in the Closure of (2,3] in the Standard Topology on the Real Numbers?

    Homework Statement Hello All, I am experiencing Adventures in Topology. So far, so good, but I have an issue here. In the topological space (Real #s, U), show that 1 is not an element of Cl((2,3]).Homework Equations The closed subsets of our topological space are the converses of the given...
  48. B

    Logical Point in Topological Problem

    Homework Statement Consider ##\mathbb{R}^\omega## in the uniform topology. Show that ##x## and ##y## lie in the same component if and only if ##x-y = (x_1-y_1,x_2-y_2,...)## is a bounded sequence. Homework Equations The uniform topology is induced by the metric ##p(x,y) := \sup_{i \in...
  49. S

    A Black hole as topological insulator

    Hi, is anyone familiar with topological insulator? I read an interesting paper: http://arxiv.org/abs/1703.09365, Black hole as topological insulator Abstract: Black holes are extraordinary massive objects which can be described classically by general relativity, and topological insulators are...