Topological , Homeomorphic version of $|S \times S|=|S| $ Give example of a subset $A$ of $\mathbb R$ such that with respect to some topology , $ A$ is homeomorphic to $A\times A$ . 
In set theory ZF it is known to be equivalent to A.C. that for any infinite set $S$ , there is a bijection between $S$ and $S \times S$ , what would be a suitable homeomorphic(topology) version of this theorem ? 
 A: The trivial examples are a one-point subspace and an infinite discrete subspace of $\Bbb R$. The simplest non-trivial examples for $A$ are the rationals, the Cantor set, the irrationals, and the Cantor set minus any one of its points: each is homeomorphic to its square. In none of these cases is this entirely trivial to prove. 


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*The Cantor set is homeomorphic to $\{0,1\}^{\Bbb N}$, where $\{0,1\}$ has the discrete topology; this is a fairly common exercise and not enormously difficult to prove. It’s easy to show that $\{0,1\}^{\Bbb N}$ is homeomorphic to its square.

*The irrationals are homeomorphic to $\Bbb N^{\Bbb N}$; this is a bit harder to prove than the fact that the Cantor set is homeomorphic to $\{0,1\}^{\Bbb N}$. However, it’s easy to show that $\Bbb N^{\Bbb N}$ is homeomorphic to its square.

*Every countable metric space without isolated points is homeomorphic to $\Bbb Q$, and $\Bbb Q^2$ is a countable metric space without isolated points, so $\Bbb Q$ is homeomorphic to its square. The characterization theorem isn’t exactly hard, but it’s certainly not trivial, either.

*The Cantor set minus a point is homeomorphic to $\Bbb N\times C$, where $C$ is the Cantor set. Thus, its square is homeomorphic to $(\Bbb N\times C)\times(\Bbb N\times C)$, which in turn is homeomorphic to $\Bbb N^2\times C^2$ and hence to $\Bbb N\times C$.

For the general question, note that if $X$ is an infinite space with the discrete topology, then $X\times X$ is homeomorphic to $X$ if and only if $|X\times X|=|X|$: any homeomorphism between $X\times X$ and $X$ is a bijection, and conversely.
