Topology Definition and 818 Threads

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?

Homework Statement Suppose that a turn of B-DNA in a circular DNA molecule with L = 100 and W = -4 becomes a turn of Z-DNA . a) What are the L, T, and W following the transition? Homework Equations L=T+W L=#bp/#bp/turn B-dna= 10.4bp/turn Z-dna=12bp/turn The Attempt at a Solution...

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...

Dear all, I recently found the topology textbooks written by Kelley, Dugundji, and Willard, which I heard that they are more concise and motivational than Munkres, which is a required text for my current topology course. I actually do not like Munkres as he is very verbose, and his problems...

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.

This September I will be going back to school after being away for 3-4 years. When I was going before, I took a class in point-set topology. I passed the class, but only with a 53. This wasn't for lack of ability, but for a lack of motivation. I dropped out of school after that semester and did...

Hi! I would like to know if my assumptions are right: Topology is the merging domain of analysis and algebra; Relational algebra use topological operators; Relational algebra is a specification of topology ?

Just for fun, I tried enumerating the topologies on n points, for small n. I found that if the space X consists of 1 point, there is only one topology, and for n = 2, there are four topologies, although two are "isomorphic" in some sense. For n = 3, I I found 26 topologies, of 7 types. For n...

I'm trying to get comfortable with the idea of a flat torus topology that is also an everywhere a smooth manifold like the video game screen where you got off the screen to the right and pop out on the left (because as I understand it this topology could be a model of space) I can't get how...

Following is from Wolfram Mathworld "A topological space, also called an abstract topological space, is a set X together with a collection of open subsets T that satisfies the four conditions: The empty set is in T.X is in T.The intersection of a finite number of sets in T is also in T.The...

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. We are going over separation axioms in class when we were asked to prove that every Urysohn Space is a Hausdorff. What I understand: A space ##X## is Urysohn space provided whenever for any...

Hello, A couple of years ago I studied abstract algebra from Dummit and Foote. However, I was never able to gain the intuition on the subject that I would like from that book. I want to study the subject again, and I want to use a different book this time around - one that covers a lot of...

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...

I am currently looking at grad schools, and I am wondering if anyone knew who are the leading researchers in differential geometry. I know that question is a little vague considering how diverse differential geometry is, but I was hoping that something could direct me in the right direction...

Homework Statement Prove that if ##(X,d)## is a metric space and ##C## and ##X \setminus C## are nonempty clopen sets, then there is an equivalent metric ##\rho## on ##X## such that ##\forall a \in C, \quad \forall b \in X \setminus C, \quad \rho(a,b) \geq 1##. I know the term "clopen" is not a...

I am trying to understand Differential Topology using several textbooks including Lee's book on Smooth Manifolds. I am looking for some good online lecture notes at undergraduate level (especially if they have good diagrams and examples) in order to supplement the texts ... Can anyone help in...