Topology Definition and 816 Threads

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. R

    Undergraduate research topics in topology?

    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...
  2. R

    New user in search of academic advice!

    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...
  3. Onyx

    A What kind of topology change does this Lorentzian metric describe?

    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...
  4. P

    I On Borel sets of the extended reals

    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...
  5. P

    I Basic question on 'bounded implies totally bounded'

    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##...
  6. P

    Show inclusion map extends to an isometry

    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...
  7. P

    I Open balls dense in closed balls in Euclidean space

    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...
  8. cianfa72

    I Show that a "cross" is not a topological manifold

    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...
  9. PhysicsRock

    I Definition of manifolds with boundary

    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...
  10. P

    I On two definitions of locally compact

    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...
  11. P

    I Verify the completion of continuous compactly supported functions

    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...
  12. cianfa72

    I Fiber bundle homeomorphism with the fiber

    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...
  13. cianfa72

    I ##SU(2, \mathbb C)## parametrization using Euler angles

    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}}...
  14. cianfa72

    I Is the Projection Restriction to a Linear Subspace a Homeomorphism?

    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##)...
  15. cianfa72

    I Is the Belt Trick Possible with Continuous Deformation in 3D Rotation Space?

    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...
  16. shiv23mj

    B Confusion with the basics of Topology (Poincare conjecture)

    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...
  17. bhobba

    Insights What Are Numbers? - Insights for Beginners

    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...
  18. P

    A Topology - Boundary of a ball without a point

    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.
  19. MidgetDwarf

    I Is the Set of Integer Outputs of sin(x) Sequentially Compact in ℝ?

    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...
  20. Infrared

    Challenge Math Challenge - June 2023

    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...
  21. cianfa72

    I Circuits or edge-disjoint unions of circuits in a connected graph

    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...
  22. A

    I Proving SL_2(C) Homeomorphic to SU(2)xT & Simple Connectedness

    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...
  23. L

    Is a Mobius Strip Truly a 2D Object in a 3D Space?

    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?
  24. 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...
  25. B

    I Questions about algebraic curves and homogeneous polynomial equations

    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...
  26. A

    A A claim about smooth maps between smooth manifolds

    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##...
  27. L

    I Create a surjective function from [0,1]^n→S^n

    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...
  28. Feynstein100

    I Does topology distinguish between real and imaginary dimensions?

    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...
  29. D

    I Are the coordinate axes a 1d- or 2d-differentiable manifold?

    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!
  30. patric44

    Standard topology is coarser than lower limit topology?

    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...
  31. Paul Colby

    I Discrete Subrings of Complex Numbers: Topology of Rings

    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...
  32. malawi_glenn

    Other Collection of Free Online Math Books and Lecture Notes (part 1)

    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...
  33. Introduction/Logic of propositions and predicates- 01 - Frederic Schuller

    Introduction/Logic of propositions and predicates- 01 - Frederic Schuller

    This is from a series of lectures - "Lectures on the Geometric Anatomy of Theoretical Physics" delivered by Dr.Frederic P Schuller
    • malawi_glenn
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    • Differential geometry Mathematical physics Topology
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    • Category: Differential Geometry
  34. B

    Does anyone know about axis-symmetric topology optimization?

    ABAQUS provides geometric restrictions such as a planer, rotational, and other symmetric, but there is no axis symmetric restriction. I know that the 2D axis symmetric element model could be possible to make in PART section. But I want to know that a full 3D element model could be optimized by...
  35. V

    I Limit cycles, differential equations and Bendixson's criterion

    I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a worked example and my doubts. The Bendixson criterion is a theorem that permits one to establish...
  36. V

    I Diverging Gaussian curvature and (non) simply connected regions

    Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...
  37. D

    Prove that if any f:X-->Y is continuous, X is the discrete topology

    Sketch of proof: ##1.## Let ##V## be open in ##Y##. ##2.## For arbitrary ##f:X\longrightarrow Y## and for arbitrary ##V##, ##f^{-1}(V)## is in ##X##. ##3.## ##f:X\longrightarrow Y## is continuous, so ##f^{-1}(V)## is open in ##X##. ##4.## Every subset ##f^{-1}(V)## of ##X## is open, so ##X##...
  38. M

    Studying Should I study Topology or Group Theory?

    Hello! I'm a physics graduate who is interested to work in Mathematical Physics. I haven't taken any specialized maths courses in undergrad, and currently I have some time to self-learn. I have finished studying Real Analysis from "Understanding Analysis - Stephen Abbott" and I'm currently...
  39. K

    B Is the Quotient Topology of Real Numbers Homeomorphic to Real Numbers?

    Hello! I have two related exercises I need help with 1. Partition the space ##\mathbb{R}## into the interval ##[a,b]##, and singletons disjoint from this interval. The associated equivalence ##\sim## is defined by ##x\sim y## if and only if either##x=y## or ##x,y\in[a,b]##. Then...
  40. K

    B Why Is the Trick for Open Sets in Quotient Topology Valid?

    Hello! Reading a textbook I found that authors use the same trick to show that subsets of quotient topology are open. And I don't understand why this trick is valid. Below I provide there example for manifold (Mobius strip) where this trick was used Quote from "Differential Geometry and...
  41. Mikaelochi

    I Doing proofs with the variety function and the Zariski topology

    I included this image because it is easier than typing it out. Anyway, this is an old problem I need to catch up on. I have a clue as to how to do part a. I could say given an x that is a member of ∩V(Ai) which implies that x is a member of V(Ai) for ∀i. Then we can say ∀i all polynomials are in...
  42. 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...
  43. Norashii

    Proof of Subspace Topology Problem: Error Identification & Explanation

    I have already seen proofs of this problem, but none of them match the one I did, therefore I would be glad if someone could indicate where is the mistake here. Thanks in advance.**My proof:** Take a limit point x of U that is not in U, but is in K (in other words x \in K \cap(\overline{U}-U))...
  44. M

    MHB Proving Topology in X: A Look at Union & Intersection

    Hey! :giggle: We consider the set $X=\mathbb{R}\cup \{\star\}$, i.e. $X$ consists of $\mathbb{R}$ and an additional point $\star$. We say that $U\subset X$ is open if: (a) For each point $x\in U\cap \mathbb{R}$ there exists an $\epsilon>0$ such that $(x-\epsilon, x+\epsilon)\subset U$...
  45. M

    MHB Convergence as for the cofinite topology on R

    Hey! :giggle: Does the sequence $x_n=\frac{1}{n}$ converges as for the cofinite topology on $\mathbb{R}$ ? If it converges,where does it converge? Could you explain to me what exactly is meant by "cofinite topology on $\mathbb{R}$" ? Do we have to define first this set and then check if we...
  46. W

    I Is [0,1] under the subspace topology Hausdorff, Compact, or Connected?

    Ok, sorry, I am being lazy here. I am tutoring intro topology and doing some refreshers. Were given the subspace topology on [0,1] generated by intervals [a,b) and I need to answer whether under this topology, [0,1] is Hausdorff, Compact or Connected. I think my solutions work , but I am looking...
  47. O

    A On the relationship between Chern number and zeros of a section

    Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the...
  48. F

    A Topology on a space of Lie algebras

    I wonder if anybody has an idea for a topology on the set of Lie algebras of a given finite dimension which is not defined via the structure constants. This condition is crucial, as I want to keep as many algebraic properties as possible, e.g. solvability, center, dimension. In the best case the...
  49. S

    Understanding the Relationship between Weak and Strong Topologies

    I do not understand what is to verify here. The problem already defined what it means to be a trivial and discrete topology but it did not state what it means to be "weak" and "strong". I assume the problem wants me to connect "weak" with trivial topology and "strong" with discrete topology, but...
  50. I

    Calculus Analysis with or without set-theoretic topology?

    Do you think a first course in analysis should focus entirely on inequalities and leave set-theoretic topology for another occasion? Should this depend on whether or not the student had a first rigorous calculus course first? If I'm not mistaken, Victor Bryant (Yet Another Introduction to...
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