# 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 $n\geq 2$ ) 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 emerges in complex dynamics. Let $f:\mathbb {C} \to \mathbb {C}$ be the complex exponential mapping defined by $f(z)=e^{z}-1$ . Let $f^{n}$ denote the $n$ -fold composition of $f$ . Then the set of all points $z\in \mathbb {C}$ such that ${\text{Im}}(f^{n}(z))\to \infty$ as $n\to \infty$ forms a collection of pairwise disjoint rays (homeomorphic copies of $[0,\infty )$ ) in the complex plane. The set of all finite endpoints of these rays is homeomorphic to $E$ . This representation can also be described as the set of all points $z\in \mathbb {C}$ such that (a) the iterates of $z$ escape to $\infty$ in the imaginary direction, and (b) $z$ is accessible via a continuous curve of points whose iterates attract to $0$ .