Topology Definition and 798 Threads

1. I have to show that S1 = {x ∈ R2 : x1 ≥ 0,x2 ≥ 0,x1 + x2 = 2} is a bounded set.2. So I have to show that sqrt(x1^2+x2^2)<M for all (x1,x2) in S1.3. I have said that M>0 and we have 0<=x1<=2 and 0<=x2<=2. And x2 = 2-x1 We can fill in sqrt(x1^2 + (2-x1)^2) = sqrt (0^2 + (2-0)^2) = 2 < M = 3...

Hello everyone, I was wondering if someone could assist me with the following problem: Let T be the topology on R generated by the topological basis B: B = {{0}, (a,b], [c,d)} a < b </ 0 0 </ c < d Compute the interior and closure of the set A: A = (−3, −2] ∪ (−1, 0) ∪ (0, 1) ∪ (2, 3) I...

I am having some trouble visualising the following problem and I hope someone will be able to help me: Let (X, dx) and (Y,dy) be metric spaces and consider their product topology X x Y (T1) and the topology T2 induced by the metric d((x1,y1),(x2,y2)) = max(dx(x1,x2),dy(y1,y2)) so the maximum of...

Hi. I'm taking a look at some lectures by Charles Kane, and he uses this simple model of polyacetylene (1D chain of atoms with alternating bonds which give alternating hopping amplitudes) [view attached image]. There are two types of polyacetylene topologically inequivalent. They both give the...

I am looking at a statement that, for a short exact sequence of Abelian groups ##0 \to A\mathop \to \limits^f B\mathop \to \limits^g C \to 0## if ##C## is a free abelian group then this short exact sequence is split I cannot figured out why, can anybody help?

Homework Statement Let ##Q = \{(x_1,x_2,...,) \in \mathbb{R}^\omega ~|~ \lim x_n = 0 \}##. I would like to show this set is closed in the uniform topology, which is generated by the metric ##\rho(x,y) = \sup d(x_i,y_i)##, where ##d## is the standard bounded metric on ##\mathbb{R}##. Homework...

Given a topological space ##(\chi, \tau)##, do mathematicians study the set of all metric functions ##d: \chi\times\chi \rightarrow [0,\infty)## that generate the topology ##\tau##? Maybe they would endow this set with additional structure too. Are there resources on this? Thanks

Homework Statement Clearly if a sequence of points ##\{x_n\}## in some space ##X## with some topology, then the sequence will also converge when ##X## is endowed with any coarser topology. I suspect this doesn't hold for endowment of ##X## with a finer topology, since a finer topology amounts...

Hi all. I would like to ask some explanation regarding how to obtain L and C values for a butterworth filter using the first Cauer topology. I will consider a normalized low pass filter. (Link added by Mentor) https://en.wikipedia.org/wiki/Butterworth_filter What I don't understand is this...

I would like to know which chapters in Munkres Topology textbook are essential for a physicist. My background in topology is limited to the topology in baby Rudin, Kreyzig's functional, and handwavy topology in intro GR books. I feel like the entire book isn't necessary, but I could be mistaken.

Has anyone seen any literature related to the construction of topological structures with geometric composition as seen below?

Hi, I'm a new member and not sure if it is ok to post a link to my blog or youtube channel. So I thought I would ask first before doing so. Is that ok or should I do it someplace else?

According to this author, http://www.math.uic.edu/undergraduate/mathclub/talks/Weeks_AMM2001.pdf, a locally Minkowski spacetime with a nontrivial global topology may have a preferred inertial frame, in the sense that hypersurfaces of constant time can only be defined using particular time...

A smooth vector field on the phase plane is known to have exactly three closed orbit. Two of the cycles, C1 and C2 lie inside the third cycle C3. However C1 does not lie inside C2, nor vice-versa. What is the configuration of the orbits? Show that there must be at least one fixed point bounded...

I am with a query about cw complex. I was thinking if is possible exist a cw complex without being of Hausdorff space. Because i was thinking that when you do a cell decomposition of a space (without being of Hausdorff) you do not obtain a 0-cell. If can exist a cw complex with space without...

Homework Statement Let A:={x∈ℝ2 : 1<x2+y2<2}. Is A open, closed or neither? Prove.Homework Equations triangle inequality d(x,y)≤d(x,z)+d(z,y) The Attempt at a Solution First I draw a picture with Wolfram Alpha. My intuition is that the set is open. Let (a,b)∈A arbitrarily and...

Hello, do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)? This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology). Thanks in advance!

Dear Physics Forum personnel, I will be taking my first graduate course (as an undergraduate) in mathematics starting on this Fall Semester. The course is about the algebraic topology (Hatcher, Spanier, Massey, etc.), which I am very excited to take as I love the topology. I am curious about...

I'm trying to prepare to read The Large Scale Structure Of Space-time by Hawking and Ellis. I've been reading a General Topology textbook since the authors say "While we expect that most of our readers will have some acquaintance with General Relativity, we have endeavored to write this book so...

I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated. ---------------------- Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...

Where can I find a free digital copy of The Large Scale Structure of Space-time by Hawking and Ellis?

I've recently been studying a bit of differential geometry in the hope of gaining a deeper understanding of the mathematics of general relativity (GR). I have come across the notion of a topology and whilst I understand the mathematical definition (in terms of endowing a set of points with the...

In your opinion what is the best book for a first approach to algebraic topology, for self studt more properly!

Okay, I am studying Baby Rudin and I am in a lot of trouble. I want to show that a closed interval [a,b] is compact in R. The book gives a proof for R^n but I am trying a different proof like thing. Since a is in some open set of an infinite open cover, the interval [a,a+r_1) is in that open set...

Hello everyone, I've 2 books on manifolds theory in e-form: 1) Spivack, calculus on manifold 2) Munkres, analysis on manifold What would be good to begin with? :oldconfused: Thank you in advance

I need some help understanding the countability and separation axioms in general topology, and how they give rise to first-countable and second-countable spaces, T1 spaces, Hausdorff spaces, etc. I more or less get the formal definition, but I can't quite grasp the intuition behind them. Any...

Let ##V## be a quaternionic vector space with quaternionic structure ##\{I,J,K\}##. One can define a Riemannian metric ##G## and hyperkahler structure ##\{\Omega^{I},\Omega^{J}, \Omega^{K}\}##. Do this inner product $$\langle p,q \rangle :=...

Hello! I want to learn about the mathematics of General Relativity, about Topology and Differential Geometry in general. I am looking for a book that has applications in physics. But, most importantly, i want a book that offers geometrical intuition(graphs and illustrations are a huge plus) but...

As a condition for a topological phase transition it seems that there must be an energy gap that closes and reopens. I have seen this many places, but never an intuitive, easy explanation. Can someone give that?

I watched this video : https://www.youtube.com/watch?v=sOiifkFYck4 Here, the lecturer said that if someone wants a spacetime which contains spin structure (physically equal to the existence of fermions, CMIIW) should topologically ℝ×Σ, where Σ is the Cauchy surface. Is that true? If so, then...

Dear Physics Forum friends, While vigorously studying Dugundji's Topology and Rudin's PMA, I found that the reference mentions the series of books written by N. Bourbaki, known as "Elements of Mathematics", and Dieudonne's Foundations of Modern Analysis. How are those books, specifically their...

Hello everyone, I'm a undergraduate at UC Berkeley. I'm doing theoretical physics but technically I'm a math major. I really want to study quantum gravity in the future. Now I have a problem of choosing courses. For next semester, I have only one spot available for either representation theory...

i: B to Y is an inclusion, p: X to Y is a covering map. Define $D=p^{-1}(B)$, we assume here B and Y are locally path-connected and semi-locally simply connected. The question 1: if B,Y, X are path-connected in what case D is path-connected (dependent on the fundamental groups)? 2 What's the...

Hello! I'm currently teaching an advanced course in mathematics at high school. The first half treats discrete mathematics, e.g. combinatorics, set theory for finite sets, and some parts of number theory. Next year I would like to change some of the subjects in the course. My question is: Are...

I was watching this video on Abstract Algebra and the professor was discussing how at one point a few mathematicians conjectured the special orthogonal group in ##\mathbb{R}^3## mod the symmetries of an icosahedron described the shape of the universe (near the end of the video). My question is...

I am studying differential geometry and I stumbled on something that I don't understand. When we have a m- dim differential manifold, with U_i and U_j open subsets of M with their corresponding coordinate function phi. As can be seen in the figure. If I understand it correctly phi_j of a...

I'm trying to understand a paper which uses weak-* topology. (Unfortunately, the paper was given to me confidentially, so I can't provide a link.) My specific question concerns a use of weak-* topology, and interpretation/use of neighborhoods in that topology. First, I'll summarize the context...

In the absence of a cosmological constant, there is a critical density (in the FLRW model) at which the universe expands asymptotically to zero velocity. If the density of the universe (without a cosmological constant) is above that critical density, at some point the expansion reverses and...

So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. When we define a manifold we require that it locally looks like Euclidean. But even the Lorentzian metric in SR does not locally looks like Euclidean let...

1. Homework Statement I'm taking a swing at Spivak's Differential Geometry, and a question that Spivak asks his reader to show is that if ##x\in M## for ##M## a manifold and there is a neighborhood (Note that Spivak requires neighborhoods to be sets which contain an open set containing the...

So proofs are a weak point of mine. The hint is that a composite of a continuous function is continuous. I'm not really sure how to use that. What I was thinking was something to the effect of an epsilon delta proof, is that applicable? Something to the effect of: ##A \sim B\text{ and let } f...

The source I'm using is: http://inperc.com/wiki/index.php?title=Homology_classes And they say Symmetry: A∼B⇒B∼A . If path q connects A to B then p connects B to A ; just pick p(t)=q(1−t),∀t . Transitivity: A∼B , B∼C⇒A∼C . If path q connects A to B and path p connects B to C then...

Hi, I am trying to prove that every convergent sequence is Cauchy - just wanted to see if my reasoning is valid and that the proof is correct. Thanks! 1. Homework Statement Prove that every convergent sequence is Cauchy Homework Equations / Theorems[/B] Theorem 1: Every convergent set is...

Homework Statement Let ε = { (-∞,a] : a∈ℝ } be the collection of all intervals of the form (-∞,a] = {x∈ℝ : x≤a} for some a∈ℝ. Is ε closed under countable unions? Homework Equations Potentially De Morgan's laws? The Attempt at a Solution Hi everyone, Thanks in advance for looking at my...

As an undergraduate, which graduate-level course will prepare me better for grad school, Complex Analysis or Topology? I probably can't fit both into my schedule, but I can definitely fit one. I have already taken undergraduate complex analysis and I'm taking now undergraduate topology. My...

Hi I have to learn some general topology within the next two months. My experience with learning is that I learn better through problem solving; 'The Fundamental Theorem of Algebra' by Fine and Rosenberger helped me a lot when I was learning abstract algebra. So, I am looking for problems that...

Quoted from Wikipedia, A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X. How to comprehend this definition in a intuitive way?

Hello, I am trying to relearn Topology. I have already read through a good amount of Munkres' book, but I was thinking of going through another. I have come across "Elementary Topology: A Problem Textbook" http://www.pdmi.ras.ru/~olegviro/topoman/e-unstable.pdf by Viro and others through another...

I am physics student.I know basic definition of topological space.I want a book(or may be any web note or video lecture) where topology spaces of various groups are rigorously discussed.