$X$ 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 :)

Solution 1:

Fix your favorite compact connected space $X$ - then $X^\omega$ is compact, connected, and homeomorphic to its own square.

Solution 2:

The Cantor set is a counter-example to the second and third statement. Note that the Cantor set is homeomorphic to $\{0,1\}^{\mathbb N}$, hence it is homeomorphic to the product with itself.

An infinite set with the smallest topology (exactly two open sets) is a counter-example to the first statement. Martini gives a better counter-example in a comment.

Solution 3:

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