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.

At this level of generality you can make $X=X \times X$ happen quite easily. Take a discrete space of any infinite cardinality, for instance. Or topologize $X=A^B$ by whatever means and compare $X \times X = A^{B \sqcup B}$; under various mild assumptions on $B$ those spaces would be homeomorphic.  
