Topology Definition and 798 Threads

I am exploring the topological reasons behind certain physical constants. In GR, is the factor of 2 in the Schwarzschild radius (Rs=2GM/c^2) ever treated as having a deeper topological significance?

This is pretty theoretical, so I don't know whether it would better belong in the "other physics" section. As I understand it, a pair of pants situation of topology change where one universe splits in two is described by a global wick-rotated riemannian metric so as to avoid the causality...

My background is calculus, but I feel that, in calculus (my textbook is Calculus, by Robert Adams), I study limits, continuity, differentiation, integrals, etc, but, I realize that if we want cool things to happen in calculus, we need some basic topological concepts: I've taken a look at metric...

Questions I have two general questions about the topic of quotient topology.. Suppose I have a set ##X## and I defined an equivalence relation ##\sim## on ##X## and I want to know what quotient toplogical sapce is hoemeorphic to. I have included a list of definitions, lemmas, propositions and...

I've a doubt regarding Lemma 1.35 (Smooth Manifold Chart Lemma) from J. Lee "Introduction to Smooth Manifolds" The proof claims that Hausdorff property follows from v). However v) includes the case where both ##p## and ##q## are included in the same ##U_{\alpha}##, i.e. their neighborhoods are...

The question might be a bit weird: which are the current "speculations" about the topology of the Universe as spacetime ? I'm aware of, from the point of view of spacelike hypersurfaces of constant cosmological time, the topology of such "spaces" might be nearly flat on large scale. What about...

if 2d = mobius strip 3d = klien bottle what could 4d be??

So, I was just doing some practice on Gauss's law, and most of the questions, when I needed to take the surface integral of something, it would be something simple, like a sphere, cylinder or at worst a torus. Though it's impractical (and probably useless) - it got me wondering, what would...

I'm a theoretical physics student working on my master's degree in China and trying to seize the opportunity to study abroad, especially in France. It's so frustrating that I still cannot find a subject or thesis that satisfies both my interests and learning goals. Here's the thing. I major in...

Hey, I was hoping for some help here. It's nothing major, I was wondering why the windings of a transformer are represented in such a manner as in the two following top-view cross-sections (I think) of the transformer. The main question is, normally it is just represented as a normal coil...

From my understanding, a topological manifold ##M## comes with, by definition, a locally euclidean topology and a (topological) atlas ##\mathcal A_1##. From this atlas one can construct the maximal atlas ##\mathcal A## throwing in all the chart maps ##(U,\varphi)## each from one of the open...

Suppose there was a bijection ##\varphi## between the 2-sphere ##M## and the euclidean plane ##\mathbb R^2##. Then one could endow ##M## with the initial topology from ##\mathbb R^2## through ##\varphi## turning it into an homeomorphism (this topology on ##M## would be different from the subset...

Consider a non-injective map ##\pi## from a set ##M## to a set ##N##. ##N## is equipped with a topological manifold structure (Hausdorff, second-countable, locally euclidean). Take the initial topology on ##M## given from ##\pi## (i.e. a set in ##M## is open iff it is the preimage under ##\pi##...

Hi, PF ##U=]2−\sqrt{2},2+\sqrt{2}[\cup{7}## is a neighborhood of ##2## because there exists an open set ##G## such that ##2\in{G}\subseteq{U}##; in this case, it might be ##G=E(2,\sqrt{2})=]2−\sqrt{2},2+\sqrt{2}[##, which is an open set, for it is an open interval, and provided it satisfies...

The annual big string theory conference took place last week. I thought I would make a thread about it, partly because the most prominent post about it anywhere, is probably Peter Woit dismissing it as worthless. I skimmed the live video (links here to slides, videos, and posters). What did I...

Let ##(f_n)## be a sequence of measurable functions from ##E## into ##\mathbb R##. I'm reading a proof of the fact that the set ##A## of all ##x\in E## for which ##f_n(x)## converges in ##\mathbb R## as ##n\to\infty## is measurable. The proof goes like this (I'm paraphrasing): Why is...

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TL;DR Summary: In search of a suitable topic for an interesting undergraduate dissertation. I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential fields and...

I am a final year Mathematics and Computing undergraduate. I am expected to submit an extensive B.Sc. thesis in four months. I have previously studied multivariable calculus, differential forms, chains, and a little bit of Theory of manifolds (Calculus on Manifolds, Michael Spivak). I am...

Looking at this paper, what sort of spatial topology change does the lorentzian metric (the first one presented) describe? Does it describe the transition from spatial connectedness to disconnectedness with time? All I know is that there is some topology change involved, but I don’t see the...

On page 45 in Folland's text on real analysis, he writes that we define Borel sets in ##\overline{\mathbb R}## by ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}##. Then he remarks that this coincides with the usual definition of...

Recall, a set ##X## is totally bounded if for each ##\epsilon>0##, there exists a finite number of open balls of radius ##\epsilon>0## that cover ##X##. Question: How can I verify that the balls ##B(\epsilon j,\epsilon)## cover ##T##? In particular, why the condition ##\epsilon |j_i|\leq 2b##...

I'm working an exercise on the completion of metric spaces. This exercise is from Gamelin and Greene's book and part of an exercise with several parts to it. I have already shown that ##\sim## is an equivalence relation, ##\rho## is a metric on ##\tilde X##, ##(\tilde X,\rho)## is complete and...

Any set with at least two elements and equipped with the discrete metric is a counterexample to the claim that the closure of an open ball is a closed ball. Yet, in the back of the back book where they present solutions to some of their exercises, they write: I feel silly for asking, but I can...

Hi, I've a doubt about the following example in "Introduction to Manifold" by L. Tu. My understanding is that if one assumes the subspace topology from ##\mathbb R^2## for the "cross", then one can show that the topological space one gets is Hausdorff, second countable but non locally...

In differential geometry, we typically define the boundary ##\partial M## of a manifold ##M## as all ##p \in M## for which there exists a chart ##(U,\varphi), p \in U## such that ##\varphi(p) \in \partial\mathbb{H}^n := \{ x \in \mathbb{R}^n : x^n = 0 \}##. Consequently, we also demand that...

I have a hard time accepting definitions that are inequivalent. So the main point of my post is to ask for confirmation that it does not matter having inequivalent definitions, but I'm not sure about this. Maybe these two definitions being inequivalent actually have some consequences. First...

Consider the attached definition of completion of a metric space. It has already been stated in my notes that ##L^p(\Omega)## equipped with ##\lVert\cdot\rVert_p## is a Banach space, hence complete. So (c) is satisfied. Also, there is a theorem that states that ##C_c(\Omega)## is a dense...

Hi, in the definition of fiber bundle there is a continuous onto map ##\pi## from the total space ##E## into the base space ##B##. Then there are local trivialization maps ##\varphi: \pi^{-1}(U) \rightarrow U \times F## where the open set ##U## in the base space is the trivializing neighborhood...

Hi, I found on some lectures the following parametrization of ##SU(2, \mathbb C)## group elements \begin{pmatrix} e^{i(\psi+\phi)/2}\cos{\frac{\theta}{2}}\ \ ie^{i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\\ ie^{-i(\psi-\phi)/2}\sin{\frac{\theta}{2}}\ \ e^{-i(\psi+\phi)/2}\cos{\frac{\theta}{2}}...

Hi, consider the Euclidean space ##\mathbb R^8## and the projection map ##\pi## over the first 4 coordinates, i.e. ##\pi : \mathbb R^8 \rightarrow \mathbb R^4##. I would show that the restriction of ##\pi## to the linear subspace ##A## (endowed with the subspace topology from ##\mathbb R^8##)...

Hi, in the following video at 15:15 the twist of ##4\pi## along the ##x## red axis is "untwisted" through a continuous deformation of the path on the sphere 3D rotations space. My concern is the following: keeping fixed the orientation in space of the start and the end of the belt, it seems...

Hi there I am trying to get into topology I am looking at the poincare conjecture if a line cannot be included as it has two fixed endpoints by the same token isn't a circle a line with two points? that has just be joined together so by the same token the circle is not allowed? Can i get a...

Hi Everyone I have been doing further investigation into infinitesimals since I wrote my insight article. I had an issue with the original article; the link to the foundations of natural numbers, integers, and rational numbers was somewhat advanced. I did need to write an insights article at...

Let us say we have f analytic in ##Ball_1(0)##. which means, radius 1, starting at ##z_0 = 0## point. If I want to find the boundary of ##Ball_1(0)##. Will the boundary be ##{0}## or ##{\emptyset}##? Not homework, just an intuition to understand ##f(z)=\frac 1 z## function ( for example ) better.

Hi. Someone showed me a problem today regarding sequentially compact sets in ℝ. Ie., is the set of the image of sin(x) and x is an integer greater than one, sequentially compact? Yes or no. What is obvious is that we know that this set is a subset of [-1,1], which is bounded. So therefore...

Welcome to the reinstatement of the monthly math challenge threads! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by...

Hi, I've a question related to the graph theory. Take a connected graph with ##n## nodes and ##b## edges. We know there are ##m = b - n + 1## fundamental circuits. Which is the total number of nonempty circuits or edge-disjoint unions of circuits ? If we do not take in account the circuit...

Using the QR decomposition (the complex version) I want to prove that ##SL_2(C)## is homeomorphic to the product ##SU(2) × T## where ##T## is the set of upper-triangular 2×2-complex matrices with real positive entries at the diagonal. Deduce that ##SL(2, C)## is simply-connect. So, I can define...

Can anyone explain the meaning behind a mobius strip? Basically just a means to travel on both sides of a flat surface? It's still a 3D object though since it uses 3D space for the twist to be possible?

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...

It is generally well-known that a plane algebraic curve is a curve in ##\mathcal{CP}^{2}## given by a homogeneous polynomial equation ##f(x,y)= \sum^{N}_{i+j=0}a_{i\,j}x^{i}y^{j}=0##, where ##i## and ##j## are nonnegative integers and not all coefficients ##a_{ij}## are zero~[1]. In addition, if...

Given the definition of a smooth map as follows: A continuous map ##f : X → Y## is smooth if for any pair of charts ##\phi : U →R^m, \psi:V →R^n## with ##U ⊂ X, V ⊂Y##, the map ##\phi(U ∩f^{-1}(V)) → R^n## given by the composition $$\psi ◦ f ◦ \phi^{-1}$$ is smooth. Claim: A map ##f : X → Y##...

the first method is this : I think I can create a surjective function f:[0,1]^n→S^n in this way : [0,1]^n is omeomorphic to D^n and D^n/S^1 is omeomorphic to S^n so finding a surjective map f is equal to finding a surjective map f':D^n →D^n/S^n and that is quotient map. Now if I take now a...

Let's say we have four 3D spaces: (x, y, z) , (x, y, iz) , (x, iy, iz) and (ix, iy, iz), with i being the imaginary unit. Now, let's say we have a donut in each of these spaces. Geometrically, the donuts are different objects, have different equations and different properties (I think) but would...

Suppose $$ D=\{ (x,0) \in \mathbb{R}^2 : x \in \mathbb{R}\} \cup \{ (0,y) \in \mathbb{R}^2 : y \in \mathbb{R} \}$$ is a subset of $$\mathbb{R}^2 $$ with subspace topology. Can this be a 1d or 2d manifold? Thank you!

Hello everyone, Our topology professor have introduced the standard topology of ##\mathbb{R}## as: $$\tau=\left\{u\subset\mathbb{R}:\forall x\in u\exists\delta>0\ s.t.\ \left(x-\delta,x+\delta\right)\subset u\right\},$$ and the lower limit topology as...

Hi, I've[1] recently become interested in discrete subrings containing 1 of the complex numbers. Being complex numbers these rings have all sorts of properties but my question may be formed in terms of the reals. The question is; when does a subring, say of the reals, ##\mathbb{R}##, becomes...

School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...

This is from a series of lectures - "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller