Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets.

Question

Prove that a proper subset $E$ of $\Bbb R^n$ is connected $\iff$ it contains exactly two relatively clopen sets.

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I researched the meaning of "clopen set". And I reached the result that so as to for a set $A$ be clopen, the set $A$ need to be both closed and open.

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I cannot do this proof. Please help me to do this. Thank you

Answer 1

I'll use the term "open" in the following as "open for the subspace topology inherited by $E$".

$\Rightarrow$: Assume that $E$ is connected. This means that the union of two disjoint non-empty open subsets of $E$ is not $E$ itself. So let's have $X$ a clopen set of $E$. $E\setminus X$ is open, then, and $E = X \cup (E\setminus X)$. What can you conclude with the remark I made above?

$\Leftarrow$: you know that $\emptyset$ and $E$ are clopen sets, so the hypothesis implies that there are no other clopen set. What can you get from this?