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**lllll** I completely overlooked that part of the question. But I did found a prove for this statement but am still lost as to how it was done, even after asking my prof.

This is what I found:

$\displaystyle U=(x-\epsilon, \ x+\epsilon), \ V = (y-\epsilon, \ y+\epsilon)$

$\displaystyle U \bigcap V = \emptyset$

suppose $\displaystyle z$ is in $\displaystyle U \bigcap V$

$\displaystyle |x-y| \leq |x-z|+|z-y| < \epsilon + \epsilon = 2\epsilon = |x-y| $

which is apparently a contradiction.

If anyone could clarify this, it would help me out a lot.