Erdős space
# 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 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 $E$.

