The quotient topology

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  • #1
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Homework Statement



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.

Homework Equations


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


The Attempt at a Solution


I thought that Uryson's metrization theorem may be used.I considered also the metric given in wikipedia.
 

Answers and Replies

  • #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.
 
  • #3
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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
 
  • #4
pasmith
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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

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.
 
  • #5
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Why should X have only finite number of connected components? -
 
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  • #6
pasmith
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Why should X have only finite number of connected components?

Re-read the definition of compactness, and recall that the collection of connected components of X is an open cover of X.
 
  • #7
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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
 
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  • #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.
 
  • #10
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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.
 
  • #11
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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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