Can a space $X$ be homeomorphic to its twofold product with itself, $X \times X$? Let $X$ be  a  topological  space  of  infinite  cardinality. Is  it  possible  for  any  $X$  to  be  homeomorphic  to  $X\times X$  $?$
For example, $\mathbb R$  is  not  homeomorphic  to  $\mathbb R^{2}$, and $S^{1}$  is not  homeomorphic  to  $S^{1} \times S^{1}$ . What  other  topological  spaces  might we consider$?$ What  properties  of  a  space  may  ensure  or  contradict  this  possibility$?$  From  the  little  topology  I  have  learnt  yet,  I  have  not  seen  this  happening.

Many Banach spaces are linearly homeomorphic to their Cartesian squares. For instance all classical spaces including $c_0$, $\ell_\infty$, $C(K)$ for $K$ compact metric, $L_p(\mu)$ for $p\in [1,\infty]$ etc. 
