# The quotient topology

## 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.

pasmith
Homework Helper
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 $\{x,y\} \subset U$".

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 $q: X \to X/\sim : x \mapsto [x]$, which in the quotient topology is continuous by definition.

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

pasmith
Homework Helper
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.

Why should X have only finite number of connected components? -

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pasmith
Homework Helper
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.

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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What have you done so far?

WannabeNewton
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.

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.

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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