What is Topology: Definition and 807 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. JasonJo

    Topology Question: Is A Open if Open Subsets of A Union to A?

    Suppose I have some subset of R, not necessarily an interval, let it be denoted as A. I have some union (might be countable, might be finite, might be uncountable) of sets where each set is an open set of A and the union of the open sets is equal to A. Can I conclude that A is open? I am not...
  2. M

    Finite-Compliment Topology and intersection of interior

    [SOLVED]Finite-Compliment Topology and intersection of interior Homework Statement Given topological space (R^{1}, finite compliment topology), find counter example to show that Arbitary Intersection of (interior of subset of R^{1}) is not equal to Interior of (arbitary intersection of...
  3. K

    How to Prove Compactness of Infinite Union of Line Segments in R^2?

    1) Prove rigorously that S={(x,y) | 1< x^2 + y^2 <4} in R^n is open using the following definition of an open set: A set S C R^n is "open" if for all x E S, there exists some r>0 s.t. all y E R^n satisfying |y-x|<r also belongs to S. [My attempt: Let x E S, r1 = 2 - |x|, r2= |x| - 1, r =...
  4. quasar987

    Checking My Topological Result: Is f^{-1}(S') \subset T?

    I just discovered the following. But since half the things I find in topology turn out to be wrong, I feel I better check with you guys. What I convinced myself of this time is that if you have a function f:(X,T)-->(Y,S) btw topological spaces, and S' is a basis for S, then to show f is...
  5. N

    Planetary topology and gravity

    Dear all, I am interested in the connection between the smoothness of a planet and the gravitational acceleration at the surface. Specifically, what is the highest a mountain can be for different values of g? More pertitently, what % of the Earth's surface would be covered with water if the...
  6. B

    Question concerning possible typo on HW (Topology)

    I'm trying to prove the following Theorem. Suppose T1 and T2 are topologies for X. The following are equivalent: 1. T1 is a subset of T2; 2. if F is closed in (X, T1), then F is closed in (X, T2); 3. if p is a limit point of A in (X, T2), then p is a limit point of A in (X, T1)...
  7. C

    Question about topology in study of electricity

    This question pertains to physics, but has to do with the math. In order to find the flux of an electric field you can put a sphere around it and use that to find flux, since the amount leaving is the same at every point. My teacher said that if you put a cube around the field/charge, you would...
  8. D

    Set theory in Munkres Topology

    In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...
  9. P

    Understanding Topology A on X: Real Numbers

    Homework Statement X is the space of all real numbers topology A={empty set} U {R} U {(-infinity,x];x in R}The Attempt at a Solution Is it because (-infinity, x] is not an open set usuing the usual metric on R but is using a metric allowed as it was not specified in the question. If not then...
  10. E

    Proof of A Dense in Rn Not Bounded in Math

    I have the following A\subset\mathbb{R}^{n} is dense then A isn't bounded. Is this true? I know that A is dense iff \bar{A}=\mathbb{R}^{n} and that A is bounded iff \exists \epsilon>0\mid B_{\epsilon}(0)\supset A. How to proof it? Or there is an counterexample?
  11. quasar987

    Exploring Compactness and Connectedness in Metric Spaces: A Topological Problem

    Homework Statement Show that if {K_n} is a decreasing family of compact connected sets in a metric space, then their intersection is connected as well. Illustrate with an example why 'compact' is necessary instead of just 'closed'. The Attempt at a Solution Well, I have a example for...
  12. quasar987

    Other ways to do this problem? (R^n topology)

    Homework Statement The problem is to show that if A is closed in R^n, and x is outside of A, then there is a point y in A such that d(y,x) = d(x,A). My method of solution involves letting b = d(x,A) = inf{d(x,z) : z in A} and considering a closed ball of radius b + delta centered on x, and...
  13. B

    Proving Triviality of pi_1(S^n;e) in Algebraic Topology

    Please read the following problem first: Suppose n > 1 and let S^n be the n-sphere in R^{n+1}. Let e be the unit-coordinate vector (1,0,...,0) on S^n. Prove that the fundamental group pi_1(S^n;e) is the trivial group. Okay, now my question is what does the notation "pi_1(S^n;e)" mean...
  14. quasar987

    Topology - is my proof correct?

    Homework Statement The lemma to prove is that "If [-R,R]^{n-1} is compact, then [-R,R]^n is too. To help us, we have two other lemmas already proven: L1: "[a,b] is compact." L2: "If A is R^n is compact and x_0 is in R^m, then A x {x_0} is compact." The Attempt at a Solution I found a proof...
  15. quasar987

    Topology - How do they know this is open?

    Homework Statement The lemma sets out to show that if A in R^n is compact and x_0 is in R^m, then A x {x_0} is compact in R^n x R^m. They say, "Let \mathcal{U} be an open cover of A x {x_0} and \mathcal{V}=\{V\subset \mathbb{R}^n:V=\{y:(y,x_0)\in U\}, \ \mbox{for some} \ U\in \mathcal{U}\}...
  16. P

    Discrete Topology: Definition & Explanation

    Defn: the discrete topology on X is defined by letting the topology consist of all open subsets of X. Why do they use the word discrete in the term discrete topology? Is it because there are subsets such that each subset contain only one point in the space. And these collection of subsets are...
  17. X

    For help, A simple question about topology

    Is a continuous 1-1 and onto mapping from the Euclidean 3-space to itself a homeomorphism? i.e. Is its inverse also continous?
  18. P

    Topological Spaces: Exploring the Existence of a Basis

    Homework Statement Does a basis exist for all sets with a topology? In other words do all topological spaces with topology T contain a basis for T?
  19. P

    Understanding Normal Topology & Examples of Non-Normal Topologies

    If a question at the end specifies wrt the normal topology, what does it mean? What would be an example of a topology that is not normal?
  20. R

    Intro Topology: boundry Q

    Homework Statement Prove that every nonempty proper subset of Rn has a nonempty boundry. The Attempt at a Solution First of all, I let S be an nonempty subset of Rn and S does not equal Rn. I tried to go about this in 2 different ways: 1) let x be in S and show that B(r,x) ∩ S ≠ ø and...
  21. T

    Topology by Munkres: Reviews & Resources

    I'm planning on buying this book, but since its so expensive I'm looking for as much information as possible on it. So you're input would be nice. Also if you have some pdf files with a chapter or so, that would really help.
  22. C

    Elementary Topology Course: Texts, Resources & Suggestions

    I have a course on this in the following year and was just wondering what kind of texts are useful for a course on elementary topology. The course description is this: "Set Theory, metric spaces and general topology. Compactness, connectedness. Urysohn's Lemma and Tietze's Theorem. Baire...
  23. I

    Where Can I Find Resources for Learning Set Theory and Topology?

    I am interested in learning set theory. It is an independent study. I already have previous knowledge of logic and deduction. Does anyone know of any good resources for learning set theory? Also, the reason I plan on learning set theory is so I can learn topology afterward, so any learning...
  24. quasar987

    Topology: is this path connected?

    Homework Statement Consider the sets A=\{(t,\sin(1/t))\in \mathbb{R}^2:t\in(0,1]\}, B=\{(0,s)\in\mathbb{R}^2:s\in[-1,1]\}. Let X=A\cup B. We consider on X the topology induced by the open ball topology of R². a) Is X connected? b) Is X path connected? The Attempt at a Solution a) I found...
  25. P

    Fundamental Group of 2-Sphere w/ 2 Disks Removed

    Homework Statement What is the fundamental group of A where A is the 2-sphere with two disjoint disks removed. It has the same homotopy type as a familiar space.Homework Equations The Attempt at a Solution When I first looked at this problem, and saw how it was drawn out (in Munkres book,) it...
  26. quasar987

    Question from last year's topology exam

    Question 1 says "Is the bouquet* of two 2-spheres a surface"? How does this question even makes sense? A surface is a paracompact Hausdorff 2-manifold w/o boundaries, and a manifold is a topological space plus an atlas. Here, no atlas is provided! *...
  27. T

    Online introduction to topology.

    Are there any good online introductions to topology?
  28. quasar987

    Topology: simple connectedness and fundamental groups

    I read on wiki* that a (pointed) topological space is simply connected iff its fundamental group is trivial. But I don't see how this in accordance with the R² caracterisation that U is simply connected iff it is path-connected and has no holes in it. Take the closed unit-disk with a point of...
  29. R

    Topology, defn of a nowhere dense set in a metric space

    Homework Statement Defn: A subset A of a metric space (X, d) is NOWHERE DENSE if its closure has empty interior. Now I am told that this implies 1. A is nowhere dense iff closure of A does not contain any non-empty open set and 2. A is nowhere dense iff each non-empty open set has a...
  30. R

    Topology, Int(A) is an open set

    Homework Statement Question: Prove Int(A) is an open set, given Int(A) is the set of all interior pts of A where x is an interior pt of A if it is the centre of an open ball in A. Homework Equations None The Attempt at a Solution Attempted Soln: Suppose x is an element of...
  31. quasar987

    A topology problem (homeomorphisms)

    Homework Statement Our professor gave us a few exercises at the end of class the other day and it is possible that he wrote the problem wrong, or I copied it wrong, but in any case, something's fishy about it. It says Show that there exists a topology on the set \mathcal{D} of all lines of...
  32. T

    Studying I'm doing EVERY exercise in munkres' topology textbook

    i think I've accelerated my learning enough, and now I'm going to start doing problems, problems, and more problems to strengthen my mathematical thinking. this thread will be devoted to munkres' well-used topology textbook. I've done all the problems in chapter 1 so far, and i haven't gotten...
  33. A

    Exploring Differential Topology: Uncovering Answers to Fundamental Questions

    I had to ask myself two simple problems in differential topology: 1) Why is the rank of a diffeomorphism (on a manifold of dimension m) of rank m? 2) Why is a chart on a manifold an embedding? These are actually quite obvious so textbooks don't even bother proving it. So I've...
  34. JasonRox

    What is the Relationship Between Metrics and Topology?

    Alright, instead of starting a new thread everytime I have a question, I will just post it in here. Note: These are not from assignments. Note: Most of these questions can be found in Topology by Munkres. I will make a mention when it is, and where it is. So, here is the first one...
  35. mattmns

    Simmons' Topology and Modern Analysis

    I am looking at an independent study next semester and I want to do something with point-set topology and analysis, so I have been looking at the book Introduction to Topology and Modern Analysis by Simmons, which I have heard good things about. For a little background I am currently taking...
  36. L

    Learn Differential Topology: Point-Set, Algebraic, & Calculus on Manifolds

    I have just found that topology is very interesting. I just want to know how one studies topology. do they go in the order of Point-set Topology, Algebraic Topology, then Differential Topology? My ultimate goal is to understand Calculus on Manifolds and Morse Theory. Is it possible to jump to...
  37. MathematicalPhysicist

    The conncetion between logic and topology.

    i think that i read that the compactness theorem in logic has a similar theorem in topology. i wanted to inquire, are there any other theorems in logic which have similar, dual theorems in topology or other branches in maths?
  38. S

    Topology of Spacetime: Non-Compactness Not Required?

    I'm familiar with the idea that there are very strong reasons to believe that possible spacetimes (M,g) for the universe can have restricted topologies. For example, I believe Hawking proved during the 70s that, given a four-dimensional manifold M and a Lorentzian metric g, then (M,g) can be...
  39. A

    Prove: Quotient Map If p:X->Y & Z Locally Compact Hausdorff Space

    Prove: If p:X->Y is a quotient map and if Z is a locally compact Hausdorff space, then the map m: p x i : X x Z -> Y x Z is a quotient map. Note: i is the identity map on Z i assume. There is a few lines of hints talking about using the tube lemma and saturated neighborhoods which i don't...
  40. MathematicalPhysicist

    Geometric Topology Vs. Algebraic Topology.

    i know that geometric topology is a field that is connected to knot theory, i wonder what are the similarities between the two subjects, and in what subject in particular they overlap?
  41. quasar987

    Topology & Physics: Motivation & Applications

    What is the motivation for a physicist to learn topology? Are there fields of physics that make explicit use of the concept of topology? (which ones) Do the ideas of topology give any insights into any topic of physics? etc.
  42. N

    Topology question; examples of non-homeomorphic metric spaces

    Hello, Here's a problem that I'm having trouble with: Give an example of metric spaces X and Y and continuous maps f: X->Y and g: Y->X such that f and g are both bijective but X and Y are not homeomorphic. I can find plenty of examples where I can find one such function, but finding...
  43. kakarukeys

    Dual Space Topology: A to B Inclusion Map

    let A \hookrightarrow B be a continuous inclusion map from A to B. A, B are two topological spaces. A \subset B what can we say about the induced map between topological dual spaces B^* \hookrightarrow A^* ? is it continuous and injective?
  44. D

    Question on Basic Topology, open sets

    Hi, In a euclidean space X with two subsets E and F, the subset E+F is defined as the collection of all x+y, where x E and y F. “+” denotes the addition in the euclidean space. Prove that if E and F are open, then E + F is open. I'd really appreciate your help. Thanks so much!
  45. A

    Is the Identity Function between Topologies Continuous?

    Let X and X' denote a single set in the two topologies T and T', respectively. Let i: X' -> X be the identity function. a) Show that i is continuous <=> T' is finer than T. Ok I am able to show that for any set in T|X this set is in T'. This is done as follows: Assume i is continuous. For...
  46. A

    Is the Definition of a Subbasis in Munkres' Topology Textbook Flawed?

    in the munkres book, they define A to be a subbasis of X if it is a collection of subsets of X whose union equals X. They define T, the topology generated by the subbasis to be the collection of all unions of finite intersections of elements of A. This definition seems to be flawed because...
  47. P

    Looking for good Topology text

    Looking for good "Topology" text I'm seeking a good text on Topology to suplement my study of geometrical methods of mathematical physics. For those of you who are learned in this field please post your favorite Topology text. Thanks. :smile: Pete
  48. A

    K topology strictly finer than standard topology

    I would like a little clarification in how to prove that the k topology on R is strictly finer than the standard topology on R. They have a proof of this in Munkres' book. I know how to prove that its finer, but the part that shows it to be strictly finer I am not sure. It says given the basis...
  49. R

    Exploring Particle-Wave Duality: A Topological Perspective

    Question. Suppose a particle {o} that is a topological entity [see:http://en.wikipedia.org/wiki/Topology] so that it can take an extended form {...o o o o o o o o o ...} to infinity. Now, suppose the transformation to exist as a wavefunction--is this then a correct view of particle-wave...
  50. JasonRox

    Topology is later today and it's an awesome professor teaching

    I walk into Real Analysis and in the first 3 seconds my thoughts go... ... drop class NOW! When I first took the course, I was under the impression that the chair of the department would be teaching the course. Unfortunately, the professor from last year Complex Analysis is teaching the...
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