Erdős space

In mathematics, Erdős space is a topological space named after Paul Erdős, who described it in 1940.[1] Erdős space is defined as a subspace ${\displaystyle E\subset \ell ^{2}}$ 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 ${\displaystyle E}$ is homeomorphic to ${\displaystyle E\times E}$ in the product topology. If the set of all homeomorphisms of the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$ (for ${\displaystyle n\geq 2}$) that leave invariant the set ${\displaystyle \mathbb {Q} ^{n}}$ of rational vectors is endowed with the compact-open topology, it becomes homeomorphic to the Erdős space.[2]

Erdős space also emerges in complex dynamics. Let ${\displaystyle f:\mathbb {C} \to \mathbb {C} }$ be the complex exponential mapping defined by ${\displaystyle f(z)=e^{z}-1}$. Let ${\displaystyle f^{n}}$ denote the ${\displaystyle n}$-fold composition of ${\displaystyle f}$. Then the set of all points ${\displaystyle z\in \mathbb {C} }$ such that ${\displaystyle {\text{Im}}(f^{n}(z))\to \infty }$ as ${\displaystyle n\to \infty }$ forms a collection of pairwise disjoint rays (homeomorphic copies of ${\displaystyle [0,\infty )}$) in the complex plane. The set of all finite endpoints of these rays is homeomorphic to ${\displaystyle E}$.[3] This representation can also be described as the set of all points ${\displaystyle z\in \mathbb {C} }$ such that (a) the iterates of ${\displaystyle z}$ escape to ${\displaystyle \infty }$ in the imaginary direction, and (b) ${\displaystyle z}$ is accessible via a continuous curve of points whose iterates attract to ${\displaystyle 0}$.