Erdős space

In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940. Erdős space is defined as a subspace $$E\subset\ell^2$$ of the Hilbert space of square summable sequences, consisting of the sequences whose elements are all rational numbers.

Erdős space is a totally disconnected, one-dimensional topological space. The space $$E$$ is homeomorphic to $$E\times E$$ in the product topology. If the set of all homeomorphisms of the Euclidean space $$\mathbb{R}^n$$ (for $$n\ge 2$$) that leave invariant the set $$\mathbb{Q}^n$$ of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.

Erdős space also surfaces in complex dynamics via iteration of the function $$f(z)=e^z-1$$. Let $$f^n$$ denote the $$n$$-fold composition of $$f$$. The set of all points $$z\in \mathbb C$$ such that $$\text{Im}(f^n(z))\to\infty$$ is a collection of pairwise disjoint rays (homeomorphic copies of $$[0,\infty)$$), each joining an endpoint in $$\mathbb C$$ to the point at infinity. The set of finite endpoints is homeomorphic to Erdős space $$E$$.