$A$ is homeomorphic to $A\times A$ Is there any infinite topological space $A$ which is connected such that $A$ and $A\times A$ are homeomorphic? 

Yes, the easiest nice one is $\Bbb R^\mathbb{N}$ probably, in the product topology (product metric). If you want ugly spaces, the indiscrete topology on $\mathbb{N}$ also works.
In fact, for any connected space $C$, the space $C^\mathbb{N}$ will be an example.  
