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. P

    What's the difference between differential topology and algebraic topology?

    Having some knowledge of differential geometry, I want to self-study topology. Which of the two areas shall I study first? Thanks for answer!
  2. B

    Is the Countable Complement Topology a Valid Topological Space?

    Homework Statement show that T:=(A subset X |A = 0 or X\A is finite) is a topology on X, Homework Equations We need to show 3 conditions. 1: X,0 are in T 2: The union of infinite open set are in T 3: The finite intersections of open sets are open. The Attempt at a...
  3. B

    Finite Complement Topology: Why It's the Finest

    Hi Guys I was wondering if anyone knows of a good link that shows why the finite complement is a Topology? I been told it is the finest topology is this right?
  4. T

    Topology Problem: Find 2 Nonhomeomorphic Compact Spaces AX[0,1]≅BX[0,1]

    Homework Statement Find two spaces A,B compact where A and B are nonhomeomorphic but AX[0,1]\congBX[0,1] Homework Equations Definitions of homeomorphism, cardinality possiby, I have no idea where to start. The Attempt at a Solution My idea Is [0,1] and S^1, but I am not sure if the...
  5. A

    Define a new topology on the reals

    Homework Statement Verify that taking \mathbb{R}, the empty set and finite sets to be closed gives a topology. Homework Equations The Attempt at a Solution Clearly the empty set is finite as it has 0 elemnts, and so is closed. If X_i , for i= {1,...,n}, are finite sets then...
  6. A

    Verifying S1 in Quotient Topology of R with x~x+1

    Homework Statement verify that R, the reals, quotiented by the equivalence relation x~x+1 is S^1 Homework Equations The Attempt at a Solution All i can think of is to draw a unit square and identify sides like the torus, but this would be using IxI, a subset of R^2, and gives a...
  7. L

    Metric space and topology help

    Let (X,d) be a metric space. Show that if there exists a metric d' on X/~ such that d(x,y) = d'([x],[y]) for all x,y in X then ~ is the identity equivalence relation, with x~y if and only if x=y. i have: assume x=y then d(x,y)=0 and [x]=[y] which implies d'([x],[y])=0 also. now...
  8. L

    Linear Map Conditions for Defining a Map on Projective Spaces

    Let \mathbb{RP}^n= ( \mathbb{R}^{n+1} - \{ 0 \} ) / \sim where x \sim y if y=\lambda x, \lambda \neq 0 \in \mathbb{R} adn the equivalence class of x is denoted [x]. what is the necessary and sufficient condition on the linear map f : \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{m+1} for the...
  9. Z

    Does Cocountable Topology Affect Countable Local Bases?

    Homework Statement Let X be a set equipped with a topology tau1, and let tau2 be the cocountable topology in which a set V in X is an open set if V is empty or X - V has only finitely or countably many elements. Consider the topology tau consisting of all sets W in X such that for each point p...
  10. D

    Feedback topology (MOSFET Clapp oscillator)

    Homework Statement I have this schematic for a clapp oscillator (see "ClappwBias") which includes the DC biasing network. When broken up into feedback topology, I end up with something like "betanet", where the output is taken from the source of the MOSFET. Homework Equations...
  11. 5

    Do I need ODE and PDE for differential topology?

    I am a senior in mathematics studying graduate point-set topoology atm. I am thinking I want to study differential topology in graduate school and maybe apply it to problems in cosmology. Do I need to take more ODE and PDE? I took intro to diff eq- the one that all engineering undergrads take...
  12. J

    What are the prerequisites for topology?

    Is calculus enough?
  13. I

    Standard topology and discrete topology

    How to compare the topology on R generated by the subbasis S={[x,y)|x,y are rational}U{(x,y]|x,y rational} to the discrete topology on R?
  14. 5

    Different metrics same topology?

    different metrics... same topology? Let (X,d) be a metric space. Define d':X x X-->[0,infinity) by ... d(x,y) if d(x,y)=<1 d'(x,y)= ... 1 if d(x,y)>=1 Prove that d' is a metric a d that d a d d' define the same topology on X. This is a weird seeming metric. I am not sure...
  15. R

    Is the Union of Intersecting Connected Sets Always Connected?

    Let A, B be two connected subsets of a topological space X such that A intersects the closure of B . Prove that A ∪ B is connected. I can prove that the union of A and the closure of B is connected, but I don't know what to do next. Could anyone give me some hints or is there another way to...
  16. T

    Topology : 3 sets on the Real line with the wada property.

    Homework Statement Find three disjoint open sets in the real line that have the same nonempty boundary. Homework Equations Connectedness on open intervals of \mathbb{R}. The Attempt at a Solution If this is it all possible, the closest thing I could come up with to 3 disjoint...
  17. tom.stoer

    Coulomb interaction and non-trivial topology

    I have a question regarding the Coulomb interaction in spaces with non-trivial topology. Suppose we have D large spatial dimensions (D>2). Then the Coulomb potential is VC(r) ~ 1/rD-2. Usually one shows in three dimensions that the Coulomb potential VC(r) is nothing else but the Fourier...
  18. R

    Defining Inside and Outside of Loops on a Closed Surface

    How do you define the inside and the outside of a loop drawn on a closed surface? For example, take a sphere. Draw a small circle around the point that is the North pole. Now you can expand the circle by pulling it down and stretching it until it fits around the equator. If you pull it down...
  19. S

    Geometric realization of topology

    Hello, Suppose that I have a cell complex and I want to define it's geometric realization, I can do it via mapping such that assign coordinates to 0-cells. however how can i do that for edges, faces and volumes. is there is ageneral formulas for lines, faces and volumes. Regards
  20. F

    Subspace/induced/relative Topology Definition

    I'm having trouble understanding the definition of a Subspace/Induced/Relative Topology. The definitions I'm finding either don't define symbols well (at all). If I understand correctly the definition is: Given: -topological space (A,\tau) -\tau={0,A,u1,u2,...un} -subset B\subsetA...
  21. J

    Simple Topology problem (Munkres)

    Hey guys, I'm reading Munkres book (2nd edition) and am caught on a problem out of Ch. 2. The problem states: If {Ta} is a family of topologies on X, show that (intersection)Ta is a topology on X. Is UTa a topology on X? Sorry for crappy notation; I don't know my way around the symbols...
  22. L

    Differential Topology: 1-dimensional manifold

    Homework Statement Given S1={(x,y) in R2: x2+y2=1}. Show that S1 is a 1-dimensional manifold. Homework Equations The Attempt at a Solution Let f1:(-1,1)->S1 s.t. f1(x)=(x,(1-x2)1/2). This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S1. I was...
  23. S

    Does A Have a Positive Eigenvalue Using Topology?

    Homework Statement Let A denote a 3x3 matrix with positive real entries. Show that A has a positive real Eigenvalue. Homework Equations This is a problem from a topology course, assigned in the chapter on fundamental groups and the Brouwer fixed point theorem.The Attempt at a Solution I...
  24. J

    Topology Math GRE Prep | Best Book Intro Guide

    I won't be able to take a course in topology before I have to take the mathematics subject test of the GRE. I have the Princeton Review guide, but I'm looking for a little stronger of a foundation. Is there a good book introductory book for this purpose? If so, what sections should I read?
  25. S

    Understanding Open and Closed Sets in Topology

    I'm self studying topology and so I don't have much direction, however I found this wonderful little pdf called topology without tears. So to get to the meat of the question, given that \tau is a topology on the set X giving (\tau,X), the members of \tau are called open sets. Up to that point...
  26. W

    Characterizing Near-Constant Functions in Discrete Product Spaces

    [Topology] Product Spaces :( Homework Statement 1. Show that in the product space N^N where the topology on N is discrete, the set of near-constant functions is dense (near constant function is a function that becomes constant from a specific index..)... 2. Prove that in R^I the set of...
  27. S

    Undergraduate Research Topics in Point-Set Topology?

    Hi all, Like the title suggests, i am interested in finding a topic in topology that would serve as the basis for a research paper. Since i am currently taking a first course in Topology (Munkres), i am basically looking for something that is not too advanced. So far i haven't been able to...
  28. M

    Proving the Connection of Subsets in Topology

    I have a question here and I'm not sure what to do as it always confuses me, any help? Let A,B be closed non-empty subsets of a topological space X with AuB and AnB connected. (i) Prove that A and B are connected. (ii) Construct disjoint non-empty disconnected subspaces A,B c R such...
  29. M

    Proof of Connectedness of A and B in Topological Space X

    Homework Statement Let A, B be closed non-empty subsets of a topological space X with A \cup B and A \cap B connected. Prove that A and B are connected. Homework Equations A set Q is not connected (disconnected) if it is expressible as a disjoint union of open sets, Q = S...
  30. H

    Any good problem book on General Topology

    I found it is hard for myself to follow the book on general topology by willard, since there are too many abstract definitions with too few examples to help me to establish these terms. I am wondering if there is any good problem book with sufficient problems that would help to make abstract...
  31. T

    What is the relationship between feedback and topology in control systems?

    Hi All, In my control systems lectures my professor talks about feedback changing the 'topology' of the system. Is he just talking about the structure of the block diagram changing, or is there some link to the topology which mathematicians refer to Regards, Thrillhouse86
  32. H

    Which algebraic topology textbook is the best for self-study?

    I see that there are four different GTM textbooks on the subject. Which one of these is the most suitable for self-study? GTM 56: Algebraic Topology: An Introduction / Massey GTM 127: A Basic Course in Algebraic Topology / Massey GTM 153: Algebraic Topology / Fulton I want to pick up...
  33. H

    Help with 2 problems about compact/pointwise convergence from Munkres - Topology

    Hi everyone, I am stuck with 2 problems from Munkres' book and I would appreciate if someone helped me solve them. Thank you in advance. Here they are: 1. Consider the sequence of continuous functions fn : ℝ -> ℝ defined by fn(x) = x/n . In which of the following three topologies does...
  34. 2

    How Do You Calculate the Homology of Spaces in Algebraic Topology?

    Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you all! Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f. i). Calculate the integral...
  35. 2

    How Do You Calculate Homology in Algebraic Topology Problems?

    Hey guys, i am studying cohomology by hatcher's. Could anyone provide me some ideas on these problems? Thank you! Let f : S2n-1 -> Sn denote a continuous map. Let Xf = D2n union f Sn be the space obtained by attaching a 2n-dim cell to Sn using the map f. i). Calculate the integral...
  36. 3

    Formulas for Cardinality & Topology of a Set

    Is there any formula that gives a relation between the cardinal number of a given set and the number of the topologies that can be taken from this set?
  37. A

    Concerns of a Math Major: Will a B+ in Topology Impact My Graduate School Apps?

    I'm a 1st semester sophomore math major right now and I'm taking an Intro Topology course this semester. It ended up being harder than I expected it to be and I'm fairly certain that I will end up with a B/B+ (most likely a B+) in the course due to one bad exam. How bad will this look when I'm...
  38. D

    Showing Regularity of X with Order Topology

    how do I show a topological space X with an order topology is regular. I've shown it is hausdorff already.
  39. J

    Analysis book that assumes some knowledge of topology

    I realize this may be kind of strange, but as it turns out, I've experienced a good introduction to topology (at the level of Munkres) before taking a rigorous class on analysis. With a month-long winter break in my near future, I'm wondering if anyone could suggest a text on real analysis that...
  40. S

    How Does a Metric Space Induce a Topology?

    When we say that a metric space (X,d) induces a topology or "every metric space is a topological space in a natural manner" we mean that: A metric space (X,d) can be seen as a topological space (X,τ) where the topology τ consists of all the open sets in the metric space? Which means that all...
  41. L

    Non-Homogeneous Topology: Finding an Example

    I'm asked to find an example of a non-homogeneous topological space. To be honest I'm not really sure where to get started. Intuitively I think I'm looking for a space where one part of it has different topological properties from another location. I just can't think of a well-known space for...
  42. M

    Prove that if X is compact and Y is Hausdorff then a continuous bijection

    Homework Statement Prove that if X is compact and Y is Hausdorff then a continuous bijection f: X \longrightarrow Y is a homeomorphism. (You may assume that a closed subspace of a compact space is compact, and that an identification space of a compact space is compact). Homework...
  43. Phrak

    Mathematica Mathematica and differential topology

    Solving equations in differential topology by pen and paper, and in Microsoft word has been tedious and error prone. Can Mathematica help? Mathematica would be required to deal with tensor equations on a pseudo Riemann manifold of four dimensions with complex matrix entries. It should be...
  44. W

    Topology question - is this function an open map? sin(1/x)

    Homework Statement This problem is from Schaum's Outline, chapter 7 #38 i believe. Let f: (0, inf) -> [-1,1] be given as f(x) = sin(1/x), where R is given the usual euclidean metric topology and (0,inf) and [-1,1] are given the relative subspace topology. Show that f is not an open map...
  45. L

    Euclidian topology ang cofinite topology

    please can you help me to prove this exercise; Prove that: the Euclidean topology R is finer than the cofinite topology on R please answer me as faster as u can I have an exame on monday and I don't know to provethis exercise!
  46. M

    Proving Discrete Topology: Topology Problem on Set of Integers

    Homework Statement Let U be a topology on the set Z of integers in which every infinite subset is open. Prove that U is the discrete topology, in which every subset is open. Homework Equations Just the definition of discrete topology The Attempt at a Solution I'm not sure where...
  47. F

    Topology: is this an open cover of an unbounded subspace of a metric space?

    Homework Statement Suppose A is an unbounded subspace of a metric space (X,d) (where d is the metric on X). Fix a point b in A let B(b,k)={a in X s.t d(b,a)<k where k>0 is a natural number}. Let A^B(b,k) denote the intersection of the subspace A with the set B(b,k). Then the...
  48. O

    Srong topology, but really a question on covering spaces

    Homework Statement Prove that the set of maps {f in Cs1(M,N)|f is a covering space} is open in the strong topology. Homework Equations The strong topology has as base neighborhoods sets of functions that are disjointly uniformly near f, along with their derivative, on compact subsets...
  49. Math Is Hard

    Bus Topology: Connecting Computers to the "Bus

    http://en.kioskea.net/contents/initiation/topologi.php3 What connects the cables coming from the computers to the "bus"? I can't quite get the idea of connecting these cables to another cable when connecting devices aren't mentioned.
  50. J

    Exploring Topology: How Can Local Theory Affect Global Structure?

    Many interesting proofs in GR (regarding black holes and singularities, etc.) involve topological methods. However, I don't understand how a theory embodied in Einstein's equations, which appear to me to be local rules of evolution, can ever change the topology since a patch of spacetime...
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