Are there any further clopen sets in $\mathbb{R}^n$ besides $\mathbb{R}^n$ and the empty set? $\mathbb{R}^n$ shall carry the topology that is induced by the canonical metric. So far, I could not find any further closed and open sets, but I cannot prove that there are no further ones.

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    $\begingroup$ Is $\mathbb R^n$ connected? $\endgroup$ – JPLF Jan 5 '14 at 15:38
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    $\begingroup$ That's a standard analysis proof (Maybe around 2nd semester) $\endgroup$ – AlexR Jan 5 '14 at 15:38

Hint: If $A\subset\mathbb R^n$ is clopen and $p\colon[0,1]\to\mathbb R^n$ is a continuous path, then the function $f\colon [0,1]\to\{0,1\}$ with $f(x)=1\iff p(x)\in A$ is continuous. Now show that the existence of a clopen $A$ which is neither empty nor the whole of $\mathbb R^n$ contradicts the intermediate value theorem.


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