Homeomorphism between topological space and product space

Question

Is there any connected topological space $X$ such that $X$ is homeomorphic to $X\times X$ ?

Answer 1

Yes, if $I$ is an infinite set and $Y$ any topological space, then $X = Y^I$ is homeomorphic to $X\times X$. If $Y$ is connected, so is $X$. A related but different example is given by the $\ell^p(\mathbb{N})$ spaces.

Answer 2

Yes. Take any infinite set equipped with the trivial topology. A singleton also has this property.

For a more interesting example, see: Hilbert cube.

Answer 3

For each natural number $i$ let $X_i=\mathbb R$ (we could choose any other connected space instead of $\mathbb R$) and define $$ X=\prod_{i\in\mathbb N} X_i. $$ Clearly $X$ is connected -- a product of connected spaces is always connected. The fact that $X\times X\cong X$ follows from the fact that, as sets, $\mathbb N\sqcup \mathbb N$ and $\mathbb N$ have the same cardinality (by $\mathbb N\sqcup \mathbb N$ I mean the union of two disjoint copies of $\mathbb N$). In fact, any bijection between $\mathbb N\sqcup \mathbb N$ and $\mathbb N$ induces a homeomorphism between $X\times X$ and $X$ by applying the bijection to indices.