Homeomorphism of topological spaces.

Question

Can we find a topological space $X$ such that $\mathbb{R}$, with usual topology is homeomorphic to $X\times X$ with product topology?

Answer 1

Let $X$ be a topological space such that $\displaystyle X\times X$ with product topology is homeomorphic to $\Bbb R$. Now $\pi:X×X\to X$ defined by $(x,y)\mapsto x$ is an surjective continuous map. Since $\Bbb R$ is connected $X×X$ is also so. And $\pi$ is continuous implies that $X$ is connected also. Now fix $y\in X$, then $X\cong X×\{y\}\subseteq X×X\cong \Bbb R$ i.e. $X$ is homeomophic to a connected subset of $\Bbb R$ i.e. $X\cong I$ for some interval $I$ of $\Bbb R$. Hence, $I\times I\cong X×X\cong\Bbb R$, which is impossible as removing one point from $I×I$ doesn't effect it's connectedness, but removing one point from $\Bbb R$ gives a disconnected set.