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. W

    What is the relationship between topology and convergence in defining open sets?

    Hello all, Sometimes I come across the situation that a topology of a space is defined indirectly through some convergence mode. I can understand when we are given a topology, we can define the convergence of a sequence w.r.t this topology. However, if we start with saying the space is...
  2. A

    Differences between Algebraic Topology and Algebraic Geometry

    i don't know if i can post it here, like this man https://www.physicsforums.com/showthread.php?t=397395, there's a lot of usefull comment for me. anyway, I'm still don't really know which one i like, either algebraic topology, or algebraic geometry. but i really do like algebra... so I'm...
  3. K

    Are Closed and Open Balls in Topology as Simple as They Seem?

    1. Is closed ball the derived set of open ball? 2. In discrete metric space, boundary of a set is always the empty set?
  4. S

    Defining Topologies: The Role of Partial Order in Point-Set Topology

    Homework Statement I started studying point-set topology a while ago, and I started to wonder, "Does a set have to be partially ordered in order to define a topology on it?" Homework Equations The Attempt at a Solution I know that every set in a topology has to be open, which...
  5. F

    Find Boundary of A (-1,1) U {2} Lower Limit Topology

    Determine the boundary of A. A= (-1,1) U {2} with the lower limit topology on R What I know is that the topology defines open sets as those of the form [a,b). In this case, if they want an interval in the form of [a,b) for the interior, then it comes to mind that [0,1) would be the...
  6. Z

    Topology- Hyperplane proof don't understand

    Homework Statement Let \mathbf {a} \in R^n be a non zero vector, and define { S = \mathbf {x} \in R^n : \mathbf {a} \cdot \mathbf {x} = 0 }. Prove that S interior = {\o} Homework Equations The Attempt at a Solution Intuitively I understand that if a is a vector in R^3...
  7. F

    Is the Countable Complement Topology a Valid Topology on the Real Line?

    Let T be the collection of subsets of R consisting of the empty set and every set whose complement is countable. a) Show that T is a topology on R. b) Show that the point 0 is a limit point of the set A= R - {0} in the countable complement topology. c) Show that in A = R -{0} there is...
  8. F

    Countable Complement Topology

    Let T be the collection of subsets of R consisting of the empty set and every set whose complement is countable. a) Show that T is a topology on R. b) Show that the point 0 is a limit point of the set A= R - {0} in the countable complement topology. c) Show that in A = R -{0} there is...
  9. N

    Find all the limit points and interior points (basic topology)

    Homework Statement Find all the limit points and interior points of following sets in R2 A={(x,y): 0<=x<=1, 0<=y<=1} *here I used "<=" symbol to name as "less then or equal". B={1-1/n: n=1,2,3,...} Homework Equations The Attempt at a Solution the limit point of B is 1 as n goes to...
  10. radou

    Continuity of a mapping in the uniform topology

    Homework Statement Let (a1, a2, ...) and (b1, b2, ...) be sequences of real numbers, where ai > 0, for every i. Let the map h : Rω --> Rω be defined with h((x1, x2, ...)) = (a1x1 + b1, a2x2 + b2, ...). One needs to investigate under what conditions on the numbers ai and bi h is continuous...
  11. radou

    Proving R^2 is metrizable in the dictionary order topology

    Homework Statement As the title suggests, I need to show that RxR is metrizable in the dictionary order topology. As a reminder, for two elements (a, b) and (c, d) of R^2, the dictionary order is defined as (a, b) < (c, d) if a < c, or if a = c and b < d. The Attempt at a Solution...
  12. F

    Closure of the Rational Numbers (Using Standard Topology)

    Prove that Cl(Q) = R in the standard topology I'm really stuck on this problem, seeing as we haven't covered limit points yet in the text and are not able to use them for this proof. Can anybody provide me with help needed for this proof? Many thanks.
  13. Z

    Is [a,b] ever an open set in the order topology?

    My textbook is indicating to me that sometimes {x \in X : a <= x <= b} is an open set. How can this happen? My only guess is that if X has a smallest and largest element, called a and b, then sure. Otherwise?
  14. Telemachus

    Topology: is this a convex set?

    Homework Statement Hi there, I have a set similar to this \{(x,y)\in{\mathbb{R}^2}:x^2+y^2\neq{k^2},k\in{\mathbb{Z}\} (its the same kind, but with elipses). And I don't know if it is convex or not. If I make the "line proof", then I should say no. What you say? Bye there, and thanks.
  15. T

    Exploring the Use of Topology in DNA Structure and Function

    I was wondering if topology has ever been utilized on the structure of DNA and how that applies to its functions? I am assuming that it has as this is one of the most obsessed over molecules in the 21st century. I am interested in this area of topology if it exists. Also I have no previous...
  16. D

    Intro Topology - Cardinality of a subset of N

    Homework Statement Every subset of \mathbb{N} is either finite or has the same cardinality as \mathbb{N} Homework Equations N/A The Attempt at a Solution Let A \subseteq \mathbb{N} and A not be finite. \mathbb{N} is countable, trivially, which means there is a bijective...
  17. radou

    Another topology on the naturals

    Homework Statement Let X = {1, 2, 3, 4, 5} and V = {{2, 3, 4}, {1, 4, 5}, {2, 4, 5}, {1, 3}} be a subbasis of a topology U on X. a) find all dense subsets of the topological space (X, U) b) let f : (X, U) --> (X, P(X)) be a mapping defined with f(x) = x (P(X)) is the partitive set of...
  18. F

    Euclidean space, euclidean topology and coordinate transformation

    Hi, I have some doubts about the precise meaning of Euclidean space. I understand Euclidean space as the metric space (\mathbb{R}^n,d) where d is the usual distance d(x,y)=\sqrt{\sum_i(x_i-y_i)^2}. Now let's supose that we have our euclidean space (in 3D for simplicity) (\mathbb{R}^3,d)...
  19. radou

    Metrizability of a certain topology

    Homework Statement Let U be the topology on R^2 whose subbase is given with the set of all lines in R^2. Is U metrizable? The Attempt at a Solution If the set of all lines (let's call it L) in R^2 is a subbase of U, then the family of all finite intersections of L forms a basis for U...
  20. G

    Find a closed interval topology

    Homework Statement Let X be an ordered set where every closed interval is compact. Prove that X has the least upper bound property. Homework Equations X having the least upper bound property means that every nonempty subset that is bounded from above has a least upper bound, in other...
  21. R

    Algebraic Topology via Categories

    Does anyone know of a modern book on algebriac topology developed in a purely categorical framework? I've been recommended Eilenberg and Steenrod (which I may end up getting regardless), but I'm looking for more recent developments in both material and pedagogy.
  22. M

    Is it a mistake to use this topology textbook for my class?

    So, the list of required texts for my fall courses came out today and I found that my topology course is requiring this piece of crap: https://www.amazon.com/dp/1441928197/?tag=pfamazon01-20. Normally I'm not scared away by bad reviews, but in this case I can't help thinking that the instructor...
  23. B

    4-dimensional topology and physics

    It is well known to mathematicians that the study of topology in 4-dimensions is more difficult than in higher dimensions due to a "lack of freedom". See for example http://hypercomplex.xpsweb.com/articles/146/en/pdf/01-09-e.pdf" Further, as mentioned in this article, some of the...
  24. B

    Learn Topology: Get Intro Book Recommendations

    Anyone would like to help me?: I started learning some mathematics in university. I would like to start learning by my own topology. Anyone have a name of a good intro. book in the area?
  25. G

    Topology and order type problem

    Homework Statement Both {1,2}x Z+ and Z+ x {1,2} are well-ordered in the dictionary order. Are they of the same order type? Why or why not? Homework Equations To be of the same order type, we must be able to construct a bijection that preserves order, that is, x<y => f(x)<f(y)...
  26. facenian

    Question about elementary topology

    Hello, I've got a simple question is the product of closed sets closed in the product topology? I think the answer is yes but need to sure
  27. P

    Topology on manifold and metric

    Is there any relation between topology on manifold (which comes from \mathbb{R}^n) and topology induced form metric in case of Remanian manifold. What if we consider pseudoremaninan manifold.
  28. facenian

    Defining the Topology of C^n: Isometry and Bijective Maps

    How is the topology in C^n defined? is it correct to think of it like this: suppose the biyective map h:C^n\rightarrow R^{2n} given by h[(z_1,\ldots,z_n)]=(x_{11},x_{12},\ldots,x_{n1},x_{n2}) where z_i=(x_{i1},x_{i2}) then the topology of C^n is defined by declaring h to be an isometry.
  29. M

    Should I Take Topology or More Analysis for My Elective Math Course?

    I'm entering into a graduate statistics program in the coming year and don't really need either class for my Master's. However, I am considering applying for a Ph.D in mathematics in the future, but for now I want to take an elective math course for fun. I've already taken a year of Real...
  30. R

    Ywhere \phi^X_t is the flow of X.

    Given a smooth manifold with no other structure (like a metric), one can define a derivative for a vector field called the Lie derivative. One can also define a Lie derivative for any tensor, including covectors. Incidentally, with antisymmetric covectors (differential forms) one can define...
  31. B

    Is U the Quotient Topology for Continuous Functions between Topological Spaces?

    Let (X; T ) be a topological space. Given the set Y and the function f : X \rightarrow Y , define U := {H\inY \mid f^{-1}(H)\in T} Show that U is the finest topology on Y with respect to which f is continuous. Homework Equations The Attempt at a Solution I was wondering is...
  32. E

    Topology Compactness Theorem: Hausdorff and Compact Topologies on a Set X

    Homework Statement Let X be a set and t & T be two topologies on X. Prove that if (X,t) is Hausdorff and (X, T) is Compact with t a subset of T, then t=T. (i.e., T is a subset of t).The Attempt at a Solution potentially useful theorem: (X,t) Hausdorff and X compact implies that each subset F...
  33. B

    Is the Set of Rational Numbers with the Relative Topology Not Locally Compact?

    Homework Statement Prove that the set of rational numbers with the relative topology as a subset of the real numbers is not locally compact Homework Equations none The Attempt at a Solution I am totally confused and want someone to give me a proof. I have looked at some stuff...
  34. C

    What is the topology generated by \EuScript{E} for X = \mathbb{R}?

    Homework Statement Let (X,\tau) be X = \mathbb{R} equipped with the topology generated by \EuScript{E} := \{[a,\infty) | a \in \mathbb{R} \}. Show that \tau = \{ \varnothing, \mathbb{R} \} \cup \{ [a,\infty), (a, \infty) | a \in \mathbb{R} \} Homework Equations A topology...
  35. Z

    Topology question - helpdrowning student

    If X is a T1, 1st countable topological space and x is a limit point of A in X, then there exists a sequence {bn} in A whose limit is x. (I'm doing this class through independent study, and in this last session the prof decided we hadn't covered enough in the semester (even though we've...
  36. Z

    Integral Calculus vs Topology vs ODE

    I'm a Physics/Math major- and am setting up my degree plan I've posted a similar thread before but now I only have one math elective left (and a boatload of choices, all of which sound interesting) I've narrowed it down to either: Integral Calculus, Topology, or Theory of Ordinary...
  37. J

    Compactness of A in R2 with Standard Topology: Tychonoff's Theorem Applied

    1. The question is to show whether A is compact in R2 with the standard topology. A = [0,1]x{0} U {1/n, n\in Z+} x [0,1] 3. If I group the [0,1] together, I get [0,1] x {0,1/n, n \in Z+ }, and [0,1] is compact in R because of Heine Borel and {0}U{1/n} is compact since you can show that every...
  38. C

    Product topology, closed subset, Hausdorff

    Homework Statement Let (X,\tau_X) and (Y,\tau_Y) be topological spaces, and let f : X \to Y be continuous. Let Y be Hausdorff, and prove that the graph of f i.e. \graph(f) := \{ (x,f(x)) | x \in X \} is a closed subset of X \times Y. Homework Equations The Attempt at a Solution...
  39. J

    Subspaces of R^n Homotopy Equivalent But Not Homeomorphic

    For each n in N give examples of subspaces of R^n, which are homotopy equivalent but NOT homeomorphic to each other. Give reasons for your answer. I'm working along the lines of open and closed intervals in R and balls in R^n with n>1. Although I'm struggling with the reasoning. Any...
  40. C

    Open sets and closed sets in product topology

    Homework Statement Let (X_a, \tau_a), a \in A be topological spaces, and let \displaystyle X = \prod_{a \in A} X_a. Homework Equations 1. Prove that the projection maps p_a : X \to X_a are open maps. 2. Let S_a \subseteq X_a and let \displaystyle S = \prod_{a \in A} S_a \subseteq...
  41. Z

    Continuity question in Topology

    Homework Statement Let (X,d) be a metric space, M a positive number, and f: X->X a continuous function for which: d(f(x), f(y)) is less than or equal to Md(x,y) for all x, y in X. Prove that f is continuous. Use this to conclude that every contractive function is continuous. The...
  42. 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!
  43. B

    Proving T is a Topology on X

    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...
  44. 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?
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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