# Topological space which is homeomorphic to its square

If $$X$$ is an infinite set, we can consider it as a topological space with the discrete topology, and it has the property that $$X$$ and $$X\times X$$ are homeomorphic. Does the property that $$X$$ is homeomorphic to $$X\times X$$ imply that $$X$$ has the discrete topology? Or are there other examples of spaces that satisfy this?

• What about the trivial topology $\{ \emptyset, X \}$? Commented Dec 31, 2019 at 13:13
• Right, I really did not think this question through enough. Anyway, good to now that there are a lot of spaces (not only those with discrete or trivial topology) that have this property. Commented Dec 31, 2019 at 13:28
• Commented Jan 9, 2020 at 15:19

For arbitrary separable infinite-dimensional Hilbert space $$H$$ you have a linear (even unitary) homeomorphism $$H \sim H \times H$$, which can be built using any countable orthonormal base in $$H$$.

Let $$Z$$ be any space and $$A$$ any infinite set. Then $$X = \prod_{\alpha \in A} Z_\alpha$$, where $$Z_\alpha = Z$$, has the property $$X \times X \approx X$$.

Spaces like that are often $$0$$ dimensional (base of clopen sets): many standard spaces are like that:

$$X=\mathbb{Q}$$ in the usual topology obeys $$X^2\simeq X$$, as both are countable metrisable spaces without isolated points.

$$X=\mathbb{P}$$ (the irrationals in the usual topology) likewise, but now because they both are separable, completely metrisable, zero-dimensional, nowhere locally compact spaces. (Or because maybe you know that $$P \simeq \mathbb{N}^\mathbb{N}$$)

$$X=C$$, the Cantor set, because $$C^2$$ and $$C$$ are both compact zero-dimensional metric spaces without isolated points.

$$X=C\setminus \{p\}$$ (for $$p \in C$$) as well, as this $$X$$ is the unique (up to homeomorphism) zero-dimensional crowded (no isolated points) metric space that is not compact, but is locally compact.

Many infinite-dimensional spaces from analysis: $$\ell^p$$ for all $$p$$, and $$\ell^\infty$$ as well and $$c_0$$ and $$c$$ (all basically topologically the same example $$\Bbb R^\mathbb{N}$$, except $$\ell^\infty$$ which is not separable).

Ugly spaces like any infinite space in the indiscrete (trivial) topology you forgot to mention.

The space can be metric and one-dimensional too: Erdős space is such a space, the irrational variant is even completely metrisable.

The Cantor space $$C = \{0,1\}^\mathbb{N}$$ clearly also has this property, but its topology isn't discrete.