What is Topological: Definition and 264 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. L

    Connected sets in a topological space

    The definition of Y being connected in a topological space (X, tau) is that you can't find two non-empty, open and disjoint sets whose union is Y. This doesn't quite make much intuitive sense to me. For example, consider R with the usual topology. Then clearly, Y= [0,1] union [2,3] is not...
  2. L

    Closed sets in a topological space

    If A\subseteq B are both subsets of a topological space (X,\tau), is it true that any closed subset of A is also a closed subset of B?
  3. J

    Degree of multiplication map of a topological group

    Hi. I'm trying to find the degree of the map of f(g,h)=g.h (i.e. multiplication in g) for fixed g. It is a map G-->G (if we fix g). We can assign a degree to this map for any topological group for which the last non-zero homology group is Z and proceed like we do for the degree of a map...
  4. T

    Topological Groups: A Guide to Properties and Solutions

    If A and B are subsets of G, let A*B denote the set of all points a*b for a in A and b in B. Let A^(-1) denote the set of all points a^(-1), for a in A. a)A neighborhood V of the identity element e is said to be symmetric if V = V^(-1) . If U is a neighborhood of e, show there is a...
  5. T

    Normal subgroup; topological group

    The integers Z are a normal subgroup of (R,+). The quotient R/Z is a familiar topological group; what is it? okay... i attempted this problem... and I don't know if i did it right... but can you guys check it? Thanks~ R/Z is a familiar topological group and Z are a normal subgroup of...
  6. T

    Show H & \bar{H} Topological Groups if H Subgroup of G

    Let H be a subspace of G. Show that if H is also a subgroup of G, then both H and\bar{H} are topological groups. So, this is what I've got... if H is a subgroup of G then H \subset G. Since H is a subspace of G then H is an open subset. But, i don't even know if that's right. How...
  7. D

    Topological properties on Linear spaces

    Is it reasonable to work with a linear space whose subspaces are considered as open subsets of the linear space when the linear space is considered as a topological Space? Actually, this linear space is spanned by a topological space with known topology.
  8. B

    Lattice QCD: topological charge

    I am reading a paper right now on lattice QCD that presents a "method that improves the cooling method and constructs an improved topological charge operator based on the product of link variables forming rectangular Wilson loops." Unfortunately I...
  9. D

    Open subspace of a compact space topological space

    It is a fact that if X is a compact topoloical space then a closed subspace of X is compact. Is an open subspace G of X also compact? please consider the following and note if i am wrong; proof: Since G is open then the relative topology on G is class {H_i}of open subset of X such that the...
  10. B

    What Is Topological Charge in Quantum Chromodynamics?

    I have been going through some papers on lattice QCD lately, and many of them mention "topological charge". I was wondering if someone could either explain what is meant by this term, or point me to a resource that has an explanation. Thanks
  11. E

    Convex Subsets of Topological Vector Spaces

    I had a quick question: Is the following proof of the theorem below correct? Theorem: If C is a convex subset of a Topological vector space X, and the origin 0 in X is contained in C, then the set tC is a subset of C for each 0<=t<=1. Proof: Since C is convex, then t*x + (1-t)*y...
  12. D

    Is C([0,1]) a Topological Vector Space?

    Let C([0,1]) be the collection of all complex-valued continuous functions on [0,1]. Define d(f,g)=\int\limits_0^{1}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}dx for all f,g \in C([0,1]) C([0,1]) is an invariant metric space. Prove that C([0,1]) is a topological vector space
  13. C

    Proving Locally Closed Property of M Union N in Topological Spaces

    Homework Statement Let X be a topological space, a subset S of X is said to be locally closed if S is the intersection of an open set and a closed set, i.e S= O intersection C where O is an open set in X and C is a closed set in X Prove that if M,N are locally closed subsets then M...
  14. S

    Let X be a topological space and let Y be a metric space

    Hello. Please, help me with this exercise: Let X be a topological space and let Y be a metric space. Let f_n: X \rightarrow Y be a sequence of continuos functions. Let x_n be a sequence of points of X converging to x. Show that if the sequence (f_n) converges uniformly to f then...
  15. M

    Topological Physics: Finding Grad School for Research

    Hey everyone, I'm an applied math undergrad whose research is in applications of topology to fluids and mechanics. I cannot find any grad school that has faculty research in topological physics. Can anybody point me anywhere? Thanks in advance...
  16. D

    Proving Banach Space Property Using Topological Isomorphism

    Homework Statement http://img219.imageshack.us/img219/2512/60637341vi6.png Homework Equations I think this is relevant: http://img505.imageshack.us/img505/336/51636887dc4.png The Attempt at a Solution A topological isomorphism implies that T and T-1 are bounded and given is that all cauchy...
  17. I

    Metric space versus Topological space

    1. Any metric space can be converted into a topological space such that an open ball in a metric space corresponds to a basis in the corresponding topology (metric spaces as a specialization of topological spaces ). 2. Any topological space can be converted into a metric space only if there is a...
  18. S

    Topological transform of singular points?

    Are singular points necessarily mapped to singular points under topological transformations? A specific example would a 2-space deformation of a triangle to any closed string with no cross over points. Would the three singular points of the triangle be necessarily mapped to three singular points...
  19. U

    Understanding Topological Space: Finding bd(0,1), cl(Q), and int(Q)

    1. Homework Statement [/b] U is T1 open iff \Re/U is countable, U=\oslash, or U=\Re. What is the bd(0,1), the cl(rationals), and the int(rationals). Homework Equations The Attempt at a Solution Could somebody please explain this problem to me? I feel like I could try it once I...
  20. F

    What is the significance of strong equivalence in topological space metrics?

    A metric dp on the topological space X×Y, with dX(x,y) and dY(x1,y1) being metrics on X and Y respectively, is defined as dp((x,y),(x1,y1))=((dX(x,y))p+(dY(x1,y1))p)1/p What does each dp((x,y),(x1,y1)) mean (geometrically or visually)? as p\rightarrow\infty...
  21. T

    What is the TOPOLOGICAL DIMENSION of?

    Let {\mathbb I} = {\mathbb R} \setminus {\mathbb Q} the set of the irrational numbers of the real line. What is the topological dimension of {\mathbb R}^2 \setminus {\mathbb I} \times {\mathbb I} ?
  22. R

    Ontological & topological issues wrt. the 'multiverse'

    Basic premiseses: Any universe can be expressed using a suitable topology containing points. At least the following (coherent) definitions exist for the multiverse (there can be more/others): 1. Topological disconnected universes Universes which could be very much the same as ours, and could...
  23. S

    What is the meaning of topological black holes?

    what is the meaning of topological black holes? thanks
  24. A

    Understanding the Smash Product of Two Topological Spaces

    Hi all, I do realize that my previous thread on CW complexes was unanswered, so perhaps I am posting my questions to wrong section of this forum. If so, please direct me to the right forum. Otherwise, I am having some problems understanding the smash product of two topological spaces. If anyone...
  25. humanino

    Perturbative Super-Yang-Mills from the Topological AdS_5xS^5 Sigma Model

    Perturbative Super-Yang-Mills from the Topological AdS_5xS^5 Sigma Model A proof of the AdS/CFT correspondence Is this serious ?
  26. L

    Topological Defects: Threads, Walls, and Rings

    1. Thread topological defects and domain walls. In which cases these defects involve new items in Hamiltonians of models? The connection between thread topological defects and first-order phase transitions (for instance in the Nielsen-Olesen model, helium II, in which Chalatnikov quantum...
  27. E

    Topological Properties of Closed Sets in the Complex Plane

    [SOLVED] Topological Properties of Closed Sets in the Complex Plane Homework Statement 1. Show that the boundary of any set D is itself a closed set. 2. Show that if D is a set and E is a closed set containing D, then E must contain the boundary of D. 3. Let C be a bounded closed convex set...
  28. A

    HELP Why CS is topological? Why BF is topological?

    Hello, I know that the CS Lagrangian is a topological invariant, in the sense that it does not depend on the connection we choose. OK, but a TFT is a field theory whose Lagrangian and all other observables do not depend on the metric, a connection in general is not uniquely defined by a metric...
  29. P

    Proof: Discreteness of Topological Groups

    Homework Statement Prove: a topological group is discrete if the singleton containing the identity is an open set. The statement is in here http://en.wikipedia.org/wiki/Discrete_group The Attempt at a Solution Is that because if you multiply the identity with any element in the group, you get...
  30. M

    Dimension of a topological space

    In Hartshorne's book definiton of a dimension is given as follows: İf X is a t.s. , dim(X) is the supremum of the integers n s.t. there exist a chain Z_0 \subsetneq Z_1...\subsetneq Z_n of distinct irreducible closed subsets of X My question is: Can we conclude directly that any...
  31. MathematicalPhysicist

    Topological continuity (a few questions).

    1.suppose that f:X->Y is continuous. if x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)? 2. suppose that f:R->R is continuous from the right, show that f is continuous when considered as a function from R_l to R, where R_l is R in the lower...
  32. marcus

    Them old quantum gravity, topological quantum field thereah blues

    J. Scott Carter from The University of Southern Alabama is a great new QG-TQFT blues performer See and listen here: http://scienceblogs.com/pontiff/2007/12/quantum_gravity_topological_qu.php Thanks to Dave Bacon! Here is the song text: The Quantum Gravity Topological Quantum Field Theory Blues...
  33. E

    Open set in a topological space

    Homework Statement If U is open set in a topological space such that U = A union B, where A and B are disjoint, do both A and B have to both be open? I think that they do not, but I cannot think of a counterexample...perhaps (-1,0) union [0,-1). OK. That's a counterexample. So, now the...
  34. N

    Countable But Not Second Countable Topological Space

    I'm wondering if someone can furnish me with either an example of a topological space that is countable (cardinality) but not second countable or a proof that countable implies second countable. Thanks.
  35. R

    Simple topological problem

    I'm really stuck on this simple problem: Let X be a topological vector space and U, V are open sets in X. Prove that U+V is open. It should be a direct consequence of the continuity of addition in topological vector spaces. But continuity states that the f^{-1}(V) is open whenever V is open...
  36. E

    Is Every Point in a Topological Space Closed?

    Homework Statement Is it true that every point in a topological space is closed? In a metric space? Homework Equations The Attempt at a Solution
  37. jal

    Topological order - string-net condensation + loop quantum gravity

    I've just read Quantum field theory of many-body systems Xiao-Gang Wen His web page http://dao.mit.edu/~wen/ I thought that his book might be easier than his papers. hehehe It's a textbook. I did get to learn a few things. Here is what wiki says about the subject. In physics...
  38. E

    Topological Bases: Are Two Definitions Equivalent?

    Homework Statement I have seen two definitions for the topology generated by a basis set: one here http://books.google.com/books?id=9cT2wI-Qrk4C&pg=PA23&dq=basis+for+a+topology&ei=wu3nRua6HYqKoQLlj81v&sig=nNFNyWXorlwCQ8WCwQBDTuI9HUc#PPA22,M1 and the other one is a set U is open if for every...
  39. A

    Study topological field theories

    hello, I have just started to study topological field theories but my problem is that I do not find of books really specialized in the subject, can somebody inform me what I must do?
  40. quasar987

    Nested set property in a general topological space

    Homework Statement The nested set property: "If {F_k} is a decreasing sequence of non empty compact sets in a metric space (M,d), then their intersection is non empty." First I cooked up a proof of my own and then I read the one provided by the book. It seemed to me that they were perfectly...
  41. P

    Finding Path Components in Topological Space X

    Homework Statement Consider a topological space X Show that every point of X is contained in a unique path component, which can be defined as the largest path connected subset of X containing this point. The Attempt at a Solution What happens if we take X=Q? There are no path connected subsets...
  42. D

    Can the reals be characterized by topological properties?

    Specifically, can they be determined (up to isomorphism of ordered fields) as the smallest connected ordered field?
  43. G

    Colimits of topological spaces

    Can someone please explain to me what the following notation/objects are: (Here X,Y are topological spaces) colim(X-->Y<--X) where the first arrow is a map f, the second is a map g. colim(X==>Y), where there are 2 maps f,g from X to Y (indicated by double lines, but couldn't draw 2 arrow...
  44. D

    Proving the Independence of Connected Sum on Open Discs: A Topological Approach

    I am trying to show that the connected sum of two topological surfaces does not depend on the open discs removed. Any hints?
  45. daniel_i_l

    Can Anyons Be Used to Create a Topological Quantum Computer?

    one of the last Scientific American issues described a way to make a QC. I wasn't sure if I understood correctly. Could someone correct me if I'm wrong: you start with pairs of anyon particles -input- and then swamp ones that are next to each other either CW or CCW. As they pass through time...
  46. Oxymoron

    Understanding the Proof of X & Y Connected Topological Spaces: A Deeper Look

    If X and Y are two connected topological spaces then so is X \otimes Y. I want to understand the proof of this theorem but I am having some difficulties. Even though we went over it in class, it is still unclear to me. The professor constructed this continuous function: f:X\otimes Y...
  47. A

    Boundaries in topological space

    I posted this on another forum, but had no response. Maybe because it's too stupid the bother with? Anyway... Say I have a set X and a topology T on X so that T = {X {} A} i.e A is an open subset of T. Then the complement of A is Ac = X - A, which is closed. Now the interior of A, int(A) is...
  48. Cincinnatus

    Visualizing topological spaces

    "visualizing" topological spaces I am taking my first topology course right now. My professor spends most of the time in class proving theorems that all sound like "if a space has property X then it must have property Y." Now this is fine, but my trouble comes in finding an example of a...
  49. Loren Booda

    Why not a quantum of any topological genus?

    Can quanta of unlimited genus exist in theory?
  50. D

    Funny definition of dimension of a topological space

    I'm reading this book (Hartshorne), and it uses a funny definition of topological dimension, which I'm having a hard time convincing myself is the usual one. The definition is as follows: dim X is the supremum of natural numbers such that there exists a chain Z_0\subset Z_1\subset \dotsb...
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