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The quotient topology

  1. Jun 26, 2013 #1
    1. The problem statement, all variables and given/known data

    X is a compact metric space, X/≈ is the quotient space,where the equivalence classes are the connected components of X.Prove that X/ ≈ is metrizable and zero dimensional.

    2. Relevant equations
    Y is zero dimensional if it has a basis consisting of clopen (closed and open at the same time)


    3. The attempt at a solution
    I thought that Uryson's metrization theorem may be used.I considered also the metric given in wikipedia.
     
  2. jcsd
  3. Jun 27, 2013 #2

    pasmith

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    Let ~ be the equivalence relation on X, which if I understand correctly is defined as "x ~ y if and only if there exists a proper clopen subset U of X such that [itex]\{x,y\} \subset U[/itex]".

    Can you establish that if X is compact, then X/~ consists of a finite number of points?

    Can you establish that the quotient topology is the discrete topology?

    You will want to consider the quotient map [itex]q: X \to X/\sim : x \mapsto [x][/itex], which in the quotient topology is continuous by definition.
     
  4. Jun 27, 2013 #3
    x~ y if and only if x and y belong to a connected set in X.S the equivalence classes ate the connected components in X
     
  5. Jun 27, 2013 #4

    pasmith

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    Yes - so x ~ y if and only if there's a connected clopen subset which contains them both, and I now see that I forgot to include the "connected" requirement.

    The remainder still stands: you can show that X/~ is a finite set under the discrete topology. You can then show, as a simple consequence of the defintions, that a finite set under the discrete topology is metrizable and zero-dimensional.
     
  6. Jun 27, 2013 #5
    Why should X have only finite number of connected components? -
     
    Last edited: Jun 27, 2013
  7. Jun 27, 2013 #6

    pasmith

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    Re-read the definition of compactness, and recall that the collection of connected components of X is an open cover of X.
     
  8. Jun 27, 2013 #7
    The number of connected components need not be finite.consider the cantor set.Also for x~y why does it have to be the same set that is both connected and clopen that includes x and y?
    Thank's
     
    Last edited: Jun 27, 2013
  9. Jun 27, 2013 #8
    What have you done so far?
     
  10. Jun 27, 2013 #9

    WannabeNewton

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    When you say zero dimensional, do you mean manifold dimension? A space is zero dimensional in the manifold sense if and only if it is a countable discrete space but the cantor set is a compact space with uncountably many connected components so the quotient space you mentioned would not be countable. This is why I ask what you mean by zero dimensional.
     
  11. Jun 27, 2013 #10
    I mean that there exists a basis for the topology consisting of clopen sets.
    I tried to find such basis using the compacity and properties of connected components,so far without results.I need some hints.
     
  12. Jun 27, 2013 #11

    micromass

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    First prove that the quotient space is Hausdorff. Then it will be a compact Hausdorff space. In compact Hausdorff spaces, you know that totally disconnected is equivalent to zero-dimensional, so you can prove that it's totally disconnected.

    Then apply some metrization theorem to show it's metrizable (you will only need to show second countable).
     
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