# Topology Definition and 96 Discussions

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. ### 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...
2. ### 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!
3. ### 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...
4. ### 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...
5. ### 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
6. ### 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...
7. ### 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...
8. ### 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...
9. ### 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...
10. ### 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...
11. ### Coarsest and finest topology

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...
12. ### Constructive Proofs Munkre's Topology Ch 1 sec. 2, ex. #1:

Summary:: Subset of Codomain is Superset of Image of Preimage, and similar proof for subset of domain I was having a hard time doing the intro chapter's exercises in Munkres' Topology text when last I worked on it, and I just wanted to make sure that there's nothing betwixt analysis and...
13. ### A Closure of constant function 1 on the complex set

I'm watching this video to which discusses how to find the domain of the self-adjoint operator for momentum on a closed interval. At moment 46:46 minutes above we consider the constant function 1 $$f:[0,2\pi] \to \mathbb{C}$$ $$f(x)=1$$ The question is that: How can we show that the...
14. ### I Product space vs fiber bundle

Hi, I'm not a really mathematician...I've a doubt about the difference between a trivial example of fiber bundle and the cartesian product space. Consider the product space ## B \times F ## : from sources I read it is an example of trivial fiber bundle with ##B## as base space and ##F## the...
15. ### Continuity of a function under Euclidean topology

Homework Statement Let ##f:X\rightarrow Y## with X = Y = ##\mathbb{R}^2## an euclidean topology. ## f(x_1,x_2) =( x^2_1+x_2*sin(x_1),x^3_2-sin(e^{x_1+x_2} ) )## Is f continuous? Homework Equations f is continuous if for every open set U in Y, its pre-image ##f^{-1}(U)## is open in X. or if...
16. ### A Structure preserved by strong equivalence of metrics?

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. We say that ##d_1## and ##d_2## are equivalent if the identity map from ##(X,d_1)## to ##(X,d_2)## and its inverse are continuous. We say that they’re uniformly equivalent if the identity map and its inverse are uniformly...
17. ### A Same open sets + same bounded sets => same Cauchy sequences?

Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
18. ### I Why can't I reach every cell in a 3x3 square?

Hi, I was playing this game in which you start from any cells of a 3x3 or 5x5 square and draw a line that loops through every cell in the box. The line can go only through a vertical or horizontal side (not diagonally). When you start from certain cells (problem cells), you can't reach at...
19. ### I Topology Words: Reasons for the particular names

From Munkres, Topology: "A topology on a set X is a collection T of subsets of X having the following properties: (1) ∅ and X are in T . (2) The union of the elements of any subcollection of T is in T . (3) The intersection of the elements of any finite subcollection of T is in T . A set X for...
20. ### B Spivak's Calculus as a Prerequisite for General Topology

High school student here... Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
21. ### I How to prove that compact regions in surfaces are closed?

This is problem 4.7.11 of O'Neill's *Elementary Differential Geometry*, second edition. The hint says to use the Hausdorff axiom ("Distinct points have distinct neighborhoods") and the results of fact that a finite intersection of neighborhoods of p is again a neighborhood of p. Here is my...
22. ### Which function drawer is this one?

Have anybody seen a 3D function drawer like this one?(4:12) Since I don’t know what these 3D drawers are really called,I can’t find any.
23. ### Question about a function of sets

Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
24. ### I Topology vs Differential Geometry

Hello. I am studying Analysis on Manifolds by Munkres. My aim is to be able to study by myself Spivak's Differential Geometry books. The problems is that the proof in Analysis on Manifolds seem many times difficult to understand and I am having SERIOUS trouble picturing myself coming up with...
25. ### I Baire Category Theorem

Hi, I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space. Assume that we have the countable collection of dense open sets ##...
26. ### I Definition of a neighborhood

Hi,t I am studying topology at the moment. I have seen that some authors define the neighborhood of a point using inclusion of an open set, while others define the term as open set that contains the point. In most of the theory I have seen so far, the latter is more convenient to use. Why is...
27. ### I Proving that an action is transitive in the orbits

<Moderator's note: Moved from General Math to Differential Geometry.> Let p:E→ B be a covering space with a group of Deck transformations Δ(p). Let b2 ∈ B be a basic point. Suppose that the action of Δ(p) on p-1(b0) is transitive. Show that for all b ∈ B the action of Δ(p)on p-1(b) is also...
28. ### I Turning the square into a circle

Hello Forum, Does topology reckon the art of turning a square into a circle? I am quite new to topology and maths in general, I have only dabbled and eyed on my collection of mathbooks. I have come to a conclusion of how to turn the Square into A Circle without cutting. I wonder if I am...
29. ### Topology: Determine whether a subset is a retract of R^2

Homework Statement Let ##X=([1,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## and ##Y=((0,\infty)\times\{0\})\cup(\cup_{n=1}^{\infty}\{n\}\times[0,1])## ##a)##Find subspaces of of the euclidean plane ##\mathbb{R}^2## which are homeomorphic to the compactification with one...
30. ### Prequisites for Nakahara's Book

For anyone who is familiar with the book "Geometry, Topology and Physics" by Nakahara, what do you think are the mathematical and physics prerequisites for this book ?
31. ### Topology: Understanding open sets

Homework Statement We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this ##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## ##a)## Show that ##\tau## satisfies that axioms for...
32. ### Finding homeomorphism between topological spaces

Homework Statement show that the two topological spaces are homeomorphic. Homework Equations Two spaces are homeomorphic if there exists a continuous bijection with a continuous inverse between them The Attempt at a Solution I have tried proving that these two spaces are homeomorphic...
33. ### A Can I change topology of the physical system smoothly?

I am encountering this kind of problem in physics. The problem is like this: Some quantity ##A## is identified as a potential field of a ##U(1)## bundle on a space ##M## (usually a torus), because it transforms like this ##{A_j}(p) = {A_i}(p) + id\Lambda (p)## in the intersection between...
34. ### A Can I find a smooth vector field on the patches of a torus?

I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this then the Chern number...
35. ### I PASSED MY QUALIFIER!

Holy bleeping bleepity bleep bleep! I now will have a Master's in mathematics - and a lot of extra time (for my family of course). -Dave K
36. ### A Integration along a loop in the base space of U(1) bundles

Let ##P## be a ##U(1)## principal bundle over base space ##M##. In physics there are phenomenons related to a loop integration in ##M##, such as the Berry's phase ##\gamma = \oint_C A ## where ##C(t)## is a loop in ##M##, and ##A## is the gauge potential (pull back of connection one-form of...
37. ### I Continuity of the determinant function

This is something I seek a proof of. Theorem: Let ## \mbox{det}:\mbox{Mat}_{n\times n}(\mathbb{R}) \rightarrow \mathbb{R}## be the determinant function assigned to a general nxn matrix with real entries. Prove this mapping is continuous. My attempt. Continuity must be judged in...
38. ### I Geometry of GR v. Spin-2 Massless Graviton Interpretation

In classical general relativity, gravity is simply a curvature of space-time. But, a quantum field theory for a massless spin-2 graviton has as its classical limit, general relativity. My question is about the topology of space-time in the hypothetical quantum field theory of a massless spin-2...
39. A

### A Are all edge states topological?

Hey am new to this forum but I have a question regarding topologically protected states.. Let's suppose we have a 1D gapped system divided two to distinct regions that have different periodicity or different properties and that at the centre, where the two regions 'meet' states appear in the...
40. ### Best Written High School Physics Text Books (SAT)

Advanced Physics (Advanced Science) by Steve Adams & Jonathan Allday from OUP Oxford: and Physics (Collins Advanced Science) 3rd Edition by Kenneth Dobson from Collins Educational: http://www.amazon.com/dp/0007267495/?tag=pfamazon01-20 Does anyone know any of these books? I find them very...
41. ### Topology Willard's General Topology vs Dugundji's Topology

Hello, I have read a fair chunk of Munkres' Topology book and took a short introductory course during undergraduate, but I would like to learn point-set topology a little better. I have quite a bit of mathematical maturity, so that isn't an issue for me. I had a larger list of potential books to...
42. ### Simple open set topology question

Homework Statement [/B] Is {n} an open set? Homework Equations [/B] To use an example, for any n that is an integer, is {10} an open set, closet set, or neither? The Attempt at a Solution [/B] I say {10} is a closed set, because it has upper and lower bounds right at 10; in other words...
43. ### How to show simplicial complex is Hausdorff?

Homework Statement Prove that any simplicial complex is Hausdorff. Homework Equations The Attempt at a Solution I have proved that for any finite simplicial complex, it is metrizable and hence Hausdorff. How to show the statement for infinite case?
44. ### Topology - Quotient Spaces

Homework Statement (part of a bigger question) For ##x,y \in \mathbb{R}^n##, write ##x \sim y \iff## there exists ##M \in GL(n,\mathbb{R})## such that ##x=My##. Show that the quotient space ##\mathbb{R}\small/ \sim## consists of two elements. Homework Equations The Attempt at a Solution...
45. ### I How to understand co-vectors?

Hi, I'm trying to get a deeper understanding of some concepts required for my next semesters but, sadly, I've found there are lots of things that are quite similar to me and they are called with different names in multiple fields of mathematics so I'm getting confused rapidly and I'd appreciate...
46. ### Topology Topology book for self-study

Hi, I would like to receive suggestions regarding (general) topology textbook for self-study. I have background in real analysis, linear and abstract algebra. I am not afraid of a challenging book. Thank you!
47. ### Open set

Homework Statement I have a set I = {x from R3 : x1<1 v x1>3 v x2<0 x x3>-1} Homework Equations Open disc B (x,r) (sqrt (x-x0)^2 + (y-y0)^2) < r The Attempt at a Solution I have done, for example by x1<1, that let r = 1-x1 Then sqrt ((x-x1)^2 + (y-y1)^2) < sqrt (x-x1)^2) < r = 1-x1 So |x-x1|...
48. ### Finding the acoustic point in a valley

Hello, I have a practical problem, I'd like to find the "best" spot to hear sounds in a valley (forgive me if "acoustic point" isn't an appropriate term, I just couldn't come up with anything better and scrolling an acoustics text didn't help), or at least a non-blind spot (one which instead...
49. ### How to prove a set is a bounded set?

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...
50. ### I Interior and closure in non-Euclidean topology

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...