$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$) Question is :
Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true:


*

*$X$ is not connected.

*$X$ is not compact

*$X$ is not homeomorphic to a subset of $\mathbb{R}$

*None of the above.


I guess first two options are false.
We do have possibility that product of two connected spaces is connected.
So, $X\times X$ is connected if $X$ is connected. So I guess there is no problem.
We do have possibility that product of two compact spaces is compact.
So, $X\times X$ is compact if $X$ is compact. So I guess there is no problem.
I understand that this is not the proof to exclude first two options but I guess the chance is more for them to be false.
So, only thing I have problem with is third option.
I could do nothing for that third option..
I would be thankful if some one can help me out to clear this.
Thank you :)

Take the Cantor set. This gives a counterexample for B and C. For A, as Mike said in the comments, take the set $[0,1]^\infty$.
