Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed quotient of a zero-dimensional space may not be zero-dimensional, even if the equivalence classes are compact.  Note that in this case, if we specify that $C/\sim$ must be totally separated, then $C/\sim$ is automatically zero-dimensional, because every totally separated compact metric space is zero-dimensional.  

This question is about similar quotients of the irrationals $\mathbb P$.  

**Question.** Let $\sim$ be a closed equivalence relation on $\mathbb P$. 

 1. Can $\mathbb P/\sim$ be connected (and non-degenerate)? 
 2. If $\mathbb P/\sim$ is totally separated, then is $\mathbb P/\sim$ necessarily zero-dimensional?  
 3. Regarding parts 1 and 2, what if, additionally, we assume that $\mathbb P/\sim$ is Polish, and that each equivalence class is compact?

By the way, *totally separated* means that for every two points $x$ and $y$ in the space, there is a clopen set containing $x$ and missing $y$. *Zero-dimensional* means that the space has a  basis of clopen sets.