What is Topological spaces: Definition and 39 Discussions

In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Other spaces, such as Euclidean spaces, metric spaces and manifolds, are topological spaces with extra structures, properties or constraints.
Although very general, topological spaces are a fundamental concept used in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

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  1. T

    B Applications of topological spaces not homeomorphic to R^n in physics

    Hello. So, the question is do you know any applications of topological spaces which are not homeomorphic to R^n in physics? Motivation for the question i am making: as i think if a topological space is homeomorphic to R^n then differential calculus is allowed on it. Modern physics uses i think...
  2. Math Amateur

    I Convergence in Topological Spaces .... Singh, Example 4.1.1 .... ....

    I am reading Tej Bahadur Singh: Elements of Topology, CRC Press, 2013 ... ... and am currently focused on Chapter 4, Section 4.1: Sequences ... I need help in order to fully understand Example 4.1.1 ...Example 4.1.1 reads as follows: In the above example from Singh we read the following: "...
  3. Math Amateur

    I Definitions of Continuity in Topological Spaces ....

    I am reading Wilson A. Sutherland's book: "Introduction to Metric & Topological Spaces" (Second Edition) ... I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ... I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...
  4. Math Amateur

    MHB Compact Topological Spaces .... Stromberg, Theorem 3.36 .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 3.36 on page 102 ... ... Theorem 3.36 and its proof read as follows: In the...
  5. Math Amateur

    MHB Compact Topological Spaces .... Stromberg, Example 3.34 (c) .... ....

    I am reading Karl R. Stromberg's book: "An Introduction to Classical Real Analysis". ... ... I am focused on Chapter 3: Limits and Continuity ... ... I need help in order to fully understand an aspect of Example 3.34 (c) on page 102 ... ... Examples 3.34 (plus some relevant definitions ...)...
  6. cianfa72

    I Injective immersion that is not a smooth embedding

    Hi, I'm aware of a typical example of injective immersion that is not a topological embedding: figure 8 ##\beta: (-\pi, \pi) \to \mathbb R^2##, with ##\beta(t)=(\sin 2t,\sin t)## As explained here an-injective-immersion-that-is-not-a-topological-embedding the image of ##\beta## is compact in...
  7. DAirey

    A The Distance between two points in a hypothetical universe

    I have a hypothetical universe where the distance between two points in spacetime is defined as: $$ds^2 =−(\phi^2 t^2)dt^2+dx^2+dy^2+dz^2$$Where ##\phi## has units of ##km s^{-2}##. The space in this universe grows quadratically with time (and, as I understand it, probably isn’t Minkowski...
  8. N

    I Equivalence of Covering Maps and Quotient Maps

    I am newbie to topology and trying to understand covering maps and quotient maps. At first sight it seems the two are closely related. For example SO(3) is double covered by SU(2) and is also the quotient SU(2)/ℤ2 so the 2 maps appear to be equivalent. Likewise, for ℝ and S1. However, I...
  9. A

    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...
  10. Wendel

    B Nesting of 2-Spheres & 2-Tori in Topological Spaces

    Is it possible to have a topological space in which three 2-spheres A, B, C are such that B is in some sense nested inside A, C is nested inside B, but A is again nested in C. What about for three 2-tori in a similar manner?
  11. nightingale123

    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...
  12. nightingale123

    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...
  13. N

    A Topological Quantum Field Theory: Help reading a paper

    https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space ##Y## and a spacetime ##X## the author talks about the associated Quantum Hilbert Spaces...
  14. Wendel

    Path-connectedness for finite topological spaces

    Homework Statement I'm trying to understand the intuition behind path-connectedness and simple-connectedness in finite topological spaces. Is there a general methodology or algorithm for finding out whether a given finite topological space is path-connected? Homework Equations how can I...
  15. S

    Topological space question

    Homework Statement Hello All, I am experiencing Adventures in Topology. So far, so good, but I have an issue here. In the topological space (Real #s, U), show that 1 is not an element of Cl((2,3]).Homework Equations The closed subsets of our topological space are the converses of the given...
  16. M

    Accumulation point of a net (topological spaces)

    Homework Statement . If ##(x_{\alpha})_{\alpha \in \Lambda}## is a net, we say that ##x \in X## is an accumulation point of the net if and only if for evey ##A \in \mathcal F_x##, the set ##\{\alpha \in \Lambda : x_{\alpha} \in A\}## is cofinal in ##Lambda##. Prove that ##x## is an accumulation...
  17. S

    INverse of a function between topological spaces and continuity

    Homework Statement Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous. Homework Equations The Attempt at a Solution I really don't know how to do this. Wikipedia entry for 'base sets' redirects to Pokemon...
  18. conquest

    Glueing together normal topological spaces at a closed subset

    Hi all! My question is the following. Suppose we have two normal topological spaces X and Y and we have a continuous map from a closed subset A of X to Y. Then we can construct another topological space by "glueing together" X and Y at A and f(A). By taking the quotient space of the disjoint...
  19. I

    Defining Topological Spaces help

    Homework Statement Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ. a) The empty set and all sets which contain closed interval [0,1] as a subset. b)R and all subsets of closed interval [0,1]. c)The empty set, ℝ and all sets...
  20. L

    Measurable spaces vs. topological spaces

    Dear All, It sounds a strange question, we know that the measure theory is the modern theory while the topological spaces is the classical analysis (roughly speaking). And measure theory solves some problems in the classical analysis. My first question is that right? Second, Is every...
  21. Fredrik

    Generalizations (from metric to topological spaces)

    This is kind of a weird question. I like to think about how I would explain things to other people, and I realized that I don't know a great way to explain in general how terms defined in the context of metric spaces are generalized to the context of topological spaces. It's not at all difficult...
  22. A

    Markov Random Topological Spaces

    The Markov chain, as you know, is a sequence of random variables with the property that any two terms of the sequence X and Y are conditionally independent given any other random variable Z that is between them. This sequence (which is in fact a family, indexed by the naturals) can and has been...
  23. Rasalhague

    Notation for Maps between Topological Spaces

    I'm used to the notation f : X --> Y for a map, where X and Y are sets. I recently came across this notation for a map between topological spaces, where the second item of each pair is a topology on the first: f : (X,{t}a) --> (Y,{tb}) Is the notation to be read "f maps each element of X...
  24. S

    Proof involving topological spaces and density.

    Homework Statement Let (A,S) and (B,T) be topological spaces and let f : A -> B be a continuous function. Suppose that D is dense in A, and that (B,T) is a Hausdorff space. Show that if f is constant on D, then f is constant on A. Homework Equations D is a dense subset of (A,S) iff the...
  25. G

    Properties of Homeomorphisms between topological spaces

    Dear all, a homomorphism is a continuous 1-1 function between two topological spaces, that is invertible with continuous inverse. My question is as follows. Let's take the topologies of two topological spaces. Is there a 1-1 function between the two collections of open sets defining the...
  26. E

    Cauchy Sequences in General Topological Spaces

    "Cauchy" Sequences in General Topological Spaces Is there an equivalent of a Cauchy sequence in a general topological space? Most definitions I have seen of "sequence" in general topological spaces assume the sequence converges within the space, and say a sequence converges if for every...
  27. H

    Topological spaces and basis

    I am having trouble proving this statement. Please help as I am trying to study for my exam, which is tomorrow Prove that the standard topology on RxR is equivalent to the one generated by the basis consisting of open disks. Thanks :)
  28. R

    Definition of a homeomorphism between topological spaces

    The definition of a homeomorphism between topological spaces X, Y, is that there exists a function Y=f(X) that is continuous and whose inverse X=f-1(Y) is also continuous. Can I assume that the function f is a bijection, since inverses only exist for bijections? Also, I thought that if a...
  29. M

    Matrices as topological spaces

    I've come across this question during revision and don't really know what you would say? Any help? Regard a 2 x 2 matrix A as a topological space by considering 2x2 matrices as vectors (a,b,c,d) as a member of R4. Let GL2(R) c R4 be the subset of the 2x2 matrices A which are invertible, i.e...
  30. S

    Two topological spaces are homeomorphic

    I had the following thought/conjecture: Two topological spaces are homeomorphic iff the two topologies are isomorphic. When I say that the two topologies are isomorphic, I mean that they are both monoids (the operation is union) and there is a bijective mapping f such that f(A) U f(B) = f(A...
  31. D

    Convergence of sequences in topological spaces?

    hi I was having difficulty with this problem in the book If (1/n) is a sequence in R which points (if any) will it converge (for every open set there is an integer N such that for all n>N 1/n is in that open set) to using the following topologies (a) Discrete (b) Indiscrete (c) { A in X ...
  32. Fredrik

    Equivalent definitions of continuity (topological spaces)

    Not really homework, but a typical exercise question, so I figured it's appropriate to post it here. Homework Statement X,Y topological spaces f:X→Y x is a point in X Prove that the following two statements are equivalent: (i) f^{-1}(E) is open for every open E that contains f(x)...
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. M

    Perfect Gen. Ordered Space Embeddable in Perfect Lin. Ordered Space

    Is it true that a perfect generalized ordered space can be embedded in a perfect linearly ordered space? It is true that a perfect generalized ordered space can be embedded as a closed subset in a perfect linearly ordered space.
  38. S

    Question on Topological Spaces

    I'm a noob starting out studying differential geometry and topology. Really probably somewhere in the multivariate calculus level, but I've been trying to understand the plethora of terminology I'm encountering with this higher math. I've been reading a lot on Wikipedia.org and PlanetMath.org...
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