Topological Definition and 265 Threads

Say we have a disconnected manifold with components C1, C2, C3. (I know in the threat title I said just topological space, but I'm actually thinking of manifolds here, sorry! Not sure how to change the title) It makes intuitive sense that if we're looking at just one of the components, then...

Author: John M. Lee Title: Introduction to Topological Manifolds Amazon Link: https://www.amazon.com/dp/1441979395/?tag=pfamazon01-20 Prerequisities: Real Analysis course involving epsilon-delta and preferebly metric spaces, group theory Level: Grad students Table of Contents: Preface...

Hi, I have a problem understanding something This is a snapshot of a book I am reading Point no. 2 concerns me, because it looks to me like it contradicts itself, with "this or this" The first part says \sum_{j}V_j = \mathbb{L^2(R)} which, to me, looks completely equivavalent...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations I have no idea. The Attempt at a Solution I seriously have no idea. This is for my analysis course, and I'm not...

In trying to get the Aharonov-effect right I've found something that I'm not sure how to sort out. Briefly put my understanding of the effect is that it shows something that cannot be explained by classical physics in the sense that makes observable a classical EM global gauge transformation...

Hello, I've found in some of the articles on experimental quantum physics the term "Topological cross section" Now I'm trying to understand what is it and in particular what the difference between topological and differential cross section? Thanks in advance for suggestions on any reading...

Hi I am studying how the spin orbit interaction in certain materials can lead to topological insulator effects and realize it has something to do with the effects of the SOC on the band structure of the material (Bi2Se3), possibly due to the inversion of the valence and conduction band but I...

It would be helpful if someone could please explain topological equivalence of functions in simple words? I am working on dynamical systems and chaos theory.In the underlying material,topological equivalence has taken a more complex definition involving orbits.Please be kind enough to explain...

Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations The Attempt at a Solution I really don't know how to do this. Wikipedia entry for 'base sets' redirects to Pokemon...

Sigh. My first post. I wrote rather long message here and as I tried posting it, "you need to login" - and it vanished. :( Anyways. I have no high physics/math education but still I consider myself enthusiasts. So be gentle! :) Consider the following "my way of filling sudokus", just thinking...

Hi, I was reading the definition of dimension from the book: "Topology", Munkres, 2nd ed. Surely I don't understand, but I wonder how ℝ2 can have dimension 2. Take the open sets U_n=\{(x,y)\mid -\infty < x <\infty, n-1<y<n+1\} for every integer n. It covers the plane but its order is...

Hi, I was curious if specific symmetries (or lack thereof) in crystal structure are necessary for the formation of topological insulators. Specifically, do we require that inversion symmetry (or inversion asymmetry) be present in the lattice in order to form the TI state? Thanks, Goalie33

Hi there! Can anybody tell me, if generically any system, which is solely described by a topological field theory, resides in a topological phase? I can't find any clear notion of topological phase. Only topological phase of matter, but I mean any kind of system. Thanks for your help.

Hello, Can anyone tell me how to go about studying Topological QFT. I am fine with QFT, Fibre bundles and currently doing Cohomology from Nakahara. Should i directly start with Witten's paper or are there any more elementary review papers? Thanks.

Changes in the internal structure during a "Topological transform" Is there any field of topology which deals with the changes in internal structure of an object when it undergoes topological transform? If I'm transforming a cube into a sphere, is there any 'field of topology' which analyze the...

I'm sorry if this is in the incorrect section, but can someone please explain what topological insulators are, the quantum hall effect, how you make a topological insulator and anything else that is relevant to the topic. Thanks.

Personally, I am interested in Topological Quantum Field Theory. And now I am battling against Quantum Field Theory. I am not sure how much Quantum Field Theory is needed to do Topological Quantum Field Theory. And I am not sure what should be the mathematical pre-requisites of Topological...

Homework Statement My question refers to the paper "Topological Sigma Models" by Edward Witten, which is available on the web after a quick google search. I am not allowed to include links in my posts, yet. I want to know how to get from equation (2.14) to (2.15). We consider a theory of maps...

Hi PF, I'm trying to come to grips with the work of Alexei Kitaev on applying notions from (topological) K-theory to the task of classifying phases of topological insulators and superconductors (paper here: http://arxiv.org/pdf/0901.2686v2.pdf). Despite having plenty of citations, I've yet to...

Hi, Prove: If (X; T1) and (Y; T2) are homeomorphism topological spaces and (X; T1) is disconnected then (Y; T2) is disconnected. I think I need some help with this proof Proof: Let (X, T1) be disconnected and let f be a homeomorphism. If f(X,T1) is disconnected then there exist two...

There is a natural way to formulate Loop quantum geometry as the dynamics of line defects in a flat vacuum. Just under 2 months ago, I attended a 90 minute seminar talk on this at the UC Berkeley physics department. Unfortunately that talk is not online, but we do have an earlier talk given last...

How do I prove that a Hausdorff topological space E is finite dimensional iff it admits a precompact neighborhood of zero?

Homework Statement I'm writing a proof for my Real Analysis III class, and in one clause I claim that the intersection of my countably infinite set of intervals {En} where En=(1+1/2+1/3+1/4+...+1/n , ∞), has the property that the infinite intersection of all En's equals ∅ (This would be a...

Hi everyone, While reading about the BHZ model used to describe HgTe quantum well topological insulators, I read at many places that the effective Hamiltonian (which is a 4 x 4 matrix) can be written in block diagonal form and the lower 2x2 block can be derived from upper 2x2 block as...

hi, can someone please help me with this problem. Let T be the collection of all U subset R such that U is open using the usual metric on R.Then (R; T ) is a topological space. The topology T could also be described as all subsets U of R such that using the usual metric on R, R \ U is...

Homework Statement Show that every topological manifold is homeomorphic to some subspace of E^n, i.e., n-dimensional Euclidean space. Homework Equations A topological manifold is a Hausdorff space that are locally Euclidean, i.e., there's an n such that for each x, there's a neighborhood...

Why is it so important to claim that the topologically protected surface states are 100% spin polarized. Is there any connection between the degree of polarization and for instance transport properties, like the absent backscattering of these states at impurities?

Hi all! My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint...

Homework Statement Prove that if a topological space has a countable base, then all bases contain a subset which is a countable base Homework Equations A base is a subset of the topological space such that all open sets can be constructed from unions and finite intersections of open sets...

Is there such thing as a bounded topological space? Or does 'boundedness' only apply to metric spaces?

Homework Statement Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ. a) The empty set and all sets which contain closed interval [0,1] as a subset. b)R and all subsets of closed interval [0,1]. c)The empty set, ℝ and all sets...

Is there a method or an algorithm or a theorem or whatever that tells us a topological space is not a Hausdorff space?

Just a small (and, really, quite useless) little nugget here: In the definition of a topological space, we require that arbitrary unions and finite intersections of open sets are open. We also need that the whole space and the empty set are also open sets. However, this last condition is...

It's often been hoped that gravity is topological, eg. Witten, Xu, Gu & Wen, Rovelli. Heckman & Verlinde make a new suggestion: http://arxiv.org/abs/1112.5210 Instantons, Twistors, and Emergent Gravity "The basic idea is to view N = 4 gauge theory on S4 as an effective low energy description of...

Here is the situation I am concerned with - Consider a smooth curve g:[0,1] \to M where M is a topological manifold (I'd be happy to assume M smooth/finite dimensional if that helps). Let Im(g) be the image of [0,1] under the map g . Give Im(g) the subspace topology induced by...

Hello! It's not really a homework problem, but it should be able to help me with something. I was just wondering: if two sets are Homeomoprh (topologically), and one of them is convex, does it mean that the other one is convex as well? Thanks a lot! Tomer.

Hello, So a topological insulator can induce a magnetic field when an electric charge is near to it (I can give a reference if necessary), but the thing is, the paper interprets the origin of this magnetic field as being the hall currents on the surface of the topological insulator. Now I...

Dear All, It sounds a strange question, we know that the measure theory is the modern theory while the topological spaces is the classical analysis (roughly speaking). And measure theory solves some problems in the classical analysis. My first question is that right? Second, Is every...

I am not a Mathematician, and I've been pondering this idea for years. I will try to describe it intelligibly. Imagine a Ring. It has three "Inputs" and three "Outputs". Any of the three "Outputs" takes you to a different Ring with three Entrances and three Exits. You cannot return to the...

[PLAIN]http://img805.imageshack.us/img805/1575/photo0138d.jpg Hi, on thursday, i have exam of advance calculus and i could not solve two problem in study sheet given by İnstructor. By 9 question, i prove by add an subtract XnY to |XnYn-XY| and i have found that |Xn(Yn-Y)+...

Is discreteness a topological property? Hey guys. I'm currently in an advanced calculus course (not topology), and the only mention of topological property in my text is that it's a property that is conserved under continuity. This section is just a brief primer on compact sets and...

Hi, this is my first post here! I've been studying about topological insulators, but still I can't understand why this materials are called topological, I've read about topological analogy between the donut and the coffee mug and the smooth changes on the Hamiltonian, but I can't get the full...

This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...

What is the difference between sigma algebra and topological space topological space?also what is the meaning of algebra on a set? the definitions are very similar except that in the case of sigma algebra the union is taken to include infinite number of sets .right?

The Markov chain, as you know, is a sequence of random variables with the property that any two terms of the sequence X and Y are conditionally independent given any other random variable Z that is between them. This sequence (which is in fact a family, indexed by the naturals) can and has been...

I'm trying to follow a proof in this video, #20 in the ThoughtSpaceZero topology series. I get the first part, but have a problem with second part, which begins at 8:16. Let there by a topological space (X,T). Let x denote an arbitrary element of X. Definition 1. Topological base. A set B...

Please help somebody on this problem... When we topologically classify the defects in ordered media, we consider the character of the fundamental group of the associated order parameter space. To construct those groups, we circumscribe the line defects by circles and the point defects by...

Is my claim correct?

Hello! Could anyone help me to resolve the impasse below? Th: Let G be a topological group and H subgroup of G. If H and G/H (quotient space of G by H) are compact, then G itself is compact. Proof: Since H is compact, the the natural mapping g of G onto G/H is a closed mapping...

I am beginning to read about the topology, I met a problem puzzled me for a while. If Y is a topological space, and X\subset Y, we can make the set X to be a topological space by defining the open set for it as U\cap X, where U is an open set of Y. I would like to show that this indeed...