Given that $X$ is a topological space with infinite cardinality and $X$ is homeomorphic to $X\times X$.
(A) $X$ cannot be connected.
(B) $X$ cannot be compact.
(C) $X$ cannot be homeomorphic to a subspace of $\mathbb R$.
(D) None of the above.
If I take $\mathbb Q$ with discrete topology then $\mathbb Q$ satisfies the given condition while contradicting option (C).
Now for options (A) and (B), I know if $X$ is connected (compact) then $X\times X$ will be connected (compact).
But I cannot think of any examples of connected or compact spaces that are homeomorphic to their two-fold product.
Can anyone provide with some examples?