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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.